,transcrib]'s ,notes3 ,und]scores 80 be=e & aft] a ^w or phrase indicate italics 9 ! orig9al text4 ,equal signs 80 be=e & aft] a ^w or phrase indicate bold 9 ! orig9al text4 ,c>at symbol 80 designates a sup]script4 ,small capitals h be5 conv]t$ to ,,solid capitals4 ,illu/rations h be5 mov$ s !y d n break up p>agraphs4 ,old or antiquat$ spell+s h be5 pres]v$4 ,typographical & punctuation errors h be5 sil5tly correct$4 ,,elem5ts ,,( ,,ari?metic4 ,,by ,,augu/us ,,de ,,morgan1 ,,( ,,tr9ity ,,college1 ,,cambridge2 ,,fell[ ,,( ,,! ,,royal ,,a/ronomical ,,society1 ,,& ,,( ,,! ,,cambridge ,,philosophical ,,society2 ,,pr(essor ,,( ,,ma!matics ,,9 ,,univ]sity ,,college1 ,,london4 8,hom9is /udiosi e/ intellig]e1 quas utilitates proprie aff]at ari?metica 81 qui solidam et p]fectam doctr9am 9 ct]is philosophi p>tibus explicant4 ,quod enim vulgo dicunt1 pr9cipium esse dimidium totius1 id vel maxime 9 philosophi p>tibus conspicitur40--,,melanc?on4 8,ce n'e/ po9t p> la r\t9e qu'on e'in/ruit1 c'e/ p> sa propre rflexion2 et il e/ ess5tiel de contract] l'habitude de se r5dre raison de ce qu'on fait3 cette habitude s'acqui]t plus facilem5t qu'on ne p5se2 et une fois acquise1 elle ne se p]d plus40--,,condillac4 ,,sev5te5? ,,?\s&4 ,,london3 ,,walton ,,& ,,mab]ly1 ,,upp] ,,g[] ,,/reet1 ,,& ,,ivy ,,lane1 ,,pat]no/] ,,r[4 ,m4,,dccc4,,lviii4 ,,london3 ,,pr9t$ ,,by ,j4 ,,w]!im] ,,& ,,co41 ,,circus-,,place1 ,,f9sbury-,,circus4 ,,preface4 ,! prec$+ editions ( ?is "w 7 publi%$ 9 #ahcj1 #ahcb1 #ahce1 & #ahdj4 ,?is fif? edition diff]s f ! ?ree prec$+1 z to ! body ( ! "w1 9 no?+ :i* ne$ prev5t ! f\r1 or any two ( !m1 f be+ us$ toge!r 9 a class4 ,b x is consid]ably augm5t$ by ! addition ( elev5 new ,app5dixes1#a relat+ to matt]s on :i* x is mo/ desirable t ! advanc$ /ud5t %\ld possess in=mation4 ,! fir/ ,app5dix1 on ,computation1 & ! six?1 on ,decimal ,money1 %\ld be read & practis$ by e /ud5t ) z mu* att5tion z any "p ( ! "w4 ,! ma/]y ( ! rules = in/antane\s conv]sion ( ! usual fractions ( a p\nd /]l+ into decimal fractions1 gives ! possessor ! great] "p ( ! advantage :i* he w\ld d]ive f ! introduction ( a decimal co9age4 ,at ! "t :5 ?is "w 0 fir/ publi%$1 ! importance ( e/abli%+ ari?metic 9 ! "y m9d ^u reason & demon/ration1 0 n admitt$ by _m4 ,! case is n[ alt]$3 s*ools exi/ 9 :i* rational ari?metic is taught1 & m]e rules are made to d no m ?an _! prop] duty4 ,!re is no necessity to advocate a *ange :i* is actually 9 progress1 z ! works :i* are publi%$ e day suffici5tly %ew4 ,& my pr9cipal reason = allud+ to ! subject "h1 is m]ely to w>n ^? :o want no?+ b r\t9e1 t ?is is n ! book = _! purpose4 ,a4 ,,de ,,morgan4 ,london1 ,may #a1 #ahdf4 #a ,"s sep>ate copies ( ^! ,app5dixes are pr9t$1 = ^? :o may desire to add !m to ! =m] editions4 ,,table ,,( ,,cont5ts4 ,,book ,i4 ,,section ,,page ,i4 ,num]ation #a ,,ii4 ,addition & ,subtraction #ad ,,iii4 ,multiplication #bd ,,iv4 ,division #cd ,v4 ,fractions #ea ,,vi4 ,decimal ,fractions #fe ,,vii4 ,squ>e ,root #hi ,,viii4 ,proportion #ajj ,,ix4 ,p]mutations & ,comb9ations #aah ,,book ,,ii4 ,i4 ,weights & ,measures1 c4 #abd ,,ii4 ,rule ( ,?ree #add ,,iii4 ,int]e/1 c4 #aej ,,app5dix4 ,i4 ,on ! mode ( comput+ #afa ,,ii4 ,on v]ification by ca/+ out n9es & elev5s #aff ,,iii4 ,on scales ( notation #afh ,,iv4 ,on ! def9ition ( fractions #aga ,v4 ,on *>act]i/ics #agd ,,vi4 ,on decimal money #agf ,,vii4 ,on ! ma9 pr9ciple ( book-keep+ #ahj ,,viii4 ,on ! r$uction ( fractions to o!rs ( ne>ly equal value #aij ,,ix4 ,on "s g5]al prop]ties ( numb]s #aic ,x4 ,on comb9ations #bja ,,xi4 ,on ,horn]'s me?od ( solv+ equations #baj ,,xii4 ,rules = ! application ( ari?metic to geometry #bag ,,elem5ts ,,( ,,ari?metic4 ,,book ,i4 ,,pr9ciples ,,( ,,ari?metic4 ,,section ,i4 ,,num]ation4 #a4 ,imag9e a multitude ( objects ( ! same k9d assembl$ toge!r2 = example1 a company ( horsem54 ,"o ( ! fir/ ?+s t mu/ /rike a spectator1 al?\gh unus$ to c\nt+1 is1 t to ea* man !re is a horse4 ,n[1 ?\gh m5 & horses are ?+s p]fectly unlike1 yet1 because !re is "o ( ! fir/ k9d to e "o ( ! second1 "o man to e horse1 a new notion w be =m$ 9 ! m9d ( ! obs]v]1 :i* we express 9 words by say+ t !re is ! same numb] ( m5 z ( horses4 ,a savage1 :o _h no o!r way ( c\nt+1 might rememb] ?is numb] by tak+ a pebble = ea* man4 ,out ( a me?od z rude z ?is has sprung our sy/em ( calculation1 by ! /eps :i* are po9t$ out 9 ! foll[+ articles4 ,suppose t !re are two companies ( horsem51 & a p]son wi%es to "k 9 :i* ( !m is ! great] numb]1 & also to be able to recollect h[ _m !re are 9 ea*4 #b4 ,suppose t :ile ! fir/ company passes by1 he drops a pebble into a basket = ea* man :om he sees4 ,!re is no connexion betwe5 ! pebbles & ! horsem5 b ?is1 t = e horseman !re is a pebble2 t is1 9 common language1 ! numb] ( pebbles & ( horsem5 is ! same4 ,suppose t :ile ! second company passes1 he drops a pebble = ea* man into a second basket3 he w !n h two baskets ( pebbles1 by :i* he w be able to convey to any o!r p]son a notion ( h[ _m horsem5 !re 7 9 ea* company4 ,:5 he wi%es to "k :i* company 0 ! l>g]1 or conta9$ mo/ horsem51 he w take a pebble out ( ea* basket1 & put !m aside4 ,he w g on do+ ?is z (t5 z he c1 t is1 until "o ( ! baskets is empti$4 ,!n1 if he also f9d ! o!r basket empty1 he says t bo? companies conta9$ ! same numb] ( horsem52 if ! second basket /ill conta9 "s pebbles1 he c tell by !m h[ _m m 7 9 ! second ?an 9 ! fir/4 #c4 ,9 ?is way a savage c\ld keep an acc\nt ( any numb]s 9 :i* he 0 int]e/$4 ,he c\ld ?us regi/] 8 *ildr51 8 cattle1 or ! numb] ( summ]s & w9t]s :i* he _h se51 by means ( pebbles1 or any o!r small objects :i* c\ld be got 9 l>ge numb]s4 ,some?+ ( ?is sort is ! practice ( savage nations at ?is day1 & x has 9 "s places la/$ ev5 aft] ! inv5tion ( bett] me?ods ( reckon+4 ,at ,rome1 9 ! "t ( ! republic1 ! prtor1 "o ( ! magi/rates1 us$ to g e ye> 9 great pomp1 & drive a nail into ! door ( ! temple ( ,jupit]2 a way ( rememb]+ ! numb] ( ye>s :i* ! city _h be5 built1 :i* probably took its rise be=e ! introduction ( writ+4 #d4 ,9 process ( "t1 names w\ld be giv5 to ^? collections ( pebbles :i* are met ) mo/ frequ5tly4 ,b z long z small numb]s only 7 requir$1 ! mo/ conv5i5t way ( reckon+ !m w\ld be by means ( ! f+]s4 ,any p]son c\ld make ) 8 two h&s ! little calculations :i* w\ld be necess>y = 8 purposes1 & w\ld "n all ! diff]5t collections ( ! f+]s4 ,he w\ld ?us get words 9 8 own language answ]+ to "o1 two1 ?ree1 f\r1 five1 six1 sev51 eight1 n9e1 & t54 ,z 8 wants increas$1 he w\ld f9d x necess>y to give names to l>g] numb]s2 b "h he w\ld be /opp$ by ! imm5se quantity ( words :i* he mu/ h1 9 ord] to express all ! numb]s :i* he w\ld be oblig$ to make use (4 ,he mu/1 !n1 aft] giv+ a sep>ate "n to a few ( ! fir/ numb]s1 manage to express all o!r numb]s by means ( ^? names4 #e4 ,i n[ %ew h[ ?is has be5 done 9 our own language4 ,! ,engli% names ( numb]s h be5 =m$ f ! ,saxon3 & 9 ! foll[+ table ea* numb] aft] t5 is writt5 d[n 9 "o column1 :ile ano!r %ews its connexion ) ^? :i* h prec$$ x4 ,"o elev5 t5 & "o#b two twelve t5 & two ?ree ?irte5 t5 & ?ree f\r f\rte5 t5 & f\r five fifte5 t5 & five six sixte5 t5 & six sev5 sev5te5 t5 & sev5 eight eighte5 t5 & eight n9e n9ete5 t5 & n9e t5 tw5ty two t5s tw5ty-"o two t5s & "o fifty five t5s tw5ty-two two t5s & two sixty six t5s c4 c4 c4 c4 sev5ty sev5 t5s ?irty ?ree t5s eighty eight t5s c4 c4 n9ety n9e t5s =ty f\r t5s a hundr$ t5 t5s c4 c4 a hundr$ & "o t5 t5s & "o c4 c4 a ?\s& t5 hundr$s t5 ?\s& a hundr$ ?\s& a million t5 hundr$ ?\s& or "o ?\s& ?\s& t5 millions a hundr$ millions c4 #b ,x has be5 suppos$ t elev5 & twelve are d]iv$ f ! ,saxon = "o left & two left 7mean+1 aft] t5 is remov$72 b !re seems bett] reason to ?9k t lev5 is a ^w mean+ t51 & connect$ ) decem4 #f4 ,words1 writt5 d[n 9 ord9>y language1 w\ld v soon be too long = su* cont9ual repetition z takes place 9 calculation4 ,%ort signs w\ld !n be sub/itut$ = words2 b x w\ld be impossible to h a di/9ct sign = e numb]3 s t :5 "s few signs _h be5 *os51 x w\ld be conv5i5t to inv5t o!rs = ! re/ out ( ^? already made4 ,! signs :i* we use areas foll[3 #j #a #b #c #d #e #f #g #h #i n\ght "o two ?ree f\r five six sev5 eight n9e ,i n[ proce$ to expla9 ! way 9 :i* ^! signs are made to repres5t o!r numb]s4 #g4 ,suppose a man fir/ to hold up "o f+]1 !n two1 & s on1 until he has held up e f+]1 & suppose a numb] ( m5 to d ! same ?+4 ,x is pla9 t we may ?us di/+ui% "o numb] f ano!r1 by caus+ two diff]5t sets ( p]sons to hold up ea* a c]ta9 numb] ( f+]s1 & t we may d ?is 9 _m diff]5t ways4 ,= example1 ! numb] fifte5 might be indicat$ ei!r by fifte5 m5 ea* hold+ up "o f+]1 or by f\r m5 ea* hold+ up two f+]s & a fif? hold+ up sev51 & s on4 ,! "q is1 ( all ^! contrivances = express+ ! numb]1 :i* is ! mo/ conv5i5t8 ,9 ! *oice :i* is made = ?is purpose consi/s :at is call$ ! me?od ( num]ation4 #h4 ,i h us$ ! =ego+ explanation because x is v probable t our sy/em ( num]ation1 & almo/ e o!r :i* is us$ 9 ! _w1 sprung f ! practice ( reckon+ on ! f+]s1 :i* *ildr5 usually foll[ :5 fir/ !y beg9 to c\nt4 ,! me?od :i* ,i h describ$ is ! rude/ possible2 b1 by a little alt]ation1 a sy/em may be =m$ :i* w enable u to express enorm\s numb]s ) great ease4 #i4 ,suppose t y are go+ to c\nt "s l>ge numb]1 = example1 to measure a numb] ( y>ds ( clo?4 ,opposite to y\rself suppose a man to be plac$1 :o keeps 8 eye ^u y1 & holds up a f+] = e y>d :i* he sees y measure4 ,:5 t5 y>ds h be5 measur$ he w h held up t5 f+]s1 & w n be able to c\nt any fur!r unless he beg9 aga91 hold+ up "o f+] at ! elev5? y>d1 two at ! twelf?1 & s on4 ,b to "k h[ _m h be5 c\nt$1 y mu/ "k1 n only h[ _m f+]s he holds up1 b also h[ _m times he has begun aga94 ,y may keep ?is 9 view by plac+ ano!r man on ! "r ( ! =m]1 :o directs 8 eye t[>ds 8 companion1 & holds up "o f+] ! mom5t he p]ceives him ready to beg9 aga91 t is1 z soon z t5 y>ds h be5 measur$4 ,ea* f+] ( ! fir/ man /&s only = "o y>d1 b ea* f+] ( ! second /&s = z _m z all ! f+]s ( ! fir/ toge!r1 t is1 = t54 ,9 ?is way a hundr$ may be c\nt$1 because ! fir/ may n[ reckon 8 t5 f+]s once = ea* f+] ( ! second man1 t is1 t5 times 9 all1 & t5 t5s is "o hundr$ 7#e74c ,n[ place a ?ird man at ! "r ( ! second1 :o %all hold up a f+] :5ev] he p]ceives ! second ready to beg9 aga94 ,"o f+] ( ! ?ird man c\nts z _m z all ! t5 f+]s ( ! second1 t is1 c\nts "o hundr$4 ,9 ?is way we may proce$ until ! ?ird has all 8 f+]s ext5d$1 :i* w signify t t5 hundr$ or "o ?\s& h be5 c\nt$ 7#e74 ,a f\r? man w\ld enable u to c\nt z f> z t5 ?\s&1 a fif? z f> z "o hundr$ ?\s&1 a six? z f> z a million1 & s on4 #c ,! ref]5ces are to ! prec$+ articles4 #aj4 ,ea* new p]son plac$ himself t[>ds y\r left 9 ! rank opposite to y4 ,n[ rule columns z 9 ! next page1 & to ! "r ( !m all place 9 words ! numb] :i* y wi% to repres5t2 9 ! fir/ column on ! "r1 place ! numb] ( f+]s :i* ! fir/ man w be hold+ up :5 t numb] ( y>ds has be5 measur$4 ,9 ! next column1 place ! f+]s :i* ! second man w !n be hold+ up2 & s on4 #g?4#f?4#e?4#d?4#crd4#bnd4#a/4 ,i4 #e #g fifty-sev54 ,,ii4 #a #j #d "o hundr$ & f\r4 ,,iii4 #a #a #j "o hundr$ & t54 ,,iv4 #b #c #d #h two ?\s& ?ree hundr$ & =ty-eight4 ,v4 #a #e #i #j #f fifte5 ?\s& n9e hundr$ & six4 ,,vi4 #a #h #g #j #j #d "o hundr$ & eighty-sev5 ?\s& & f\r4 ,,vii4 #c #f #i #g #b #h #e ?ree million1 six hundr$ & n9ety-sev5 ?\s&1 two hundr$ & eighty-five4 #aa4 ,9 ,i4 ! numb] fifty-sev5 is express$4 ,?is means 7#e7 five t5s & sev54 ,! fir/ has !re=e c\nt$ all 8 f+]s five times1 & has c\nt$ sev5 f+]s m4 ,?is is %ewn by five f+]s ( ! second man be+ held up1 & sev5 ( ! fir/4 ,9 ,,ii4 ! numb] "o hundr$ & f\r is repres5t$4 ,?is numb] is 7#e7 t5 t5s & f\r4 ,! second p]son has !re=e j reckon$ all 8 f+]s once1 :i* is d5ot$ by ! ?ird p]son hold+ up "o f+]2 b he has n yet begun aga91 because he does n hold up a f+] until ! fir/ has c\nt$ t51 ( :i* t5 only f\r are complet$4 ,:5 all ! la/-m5tion$ t5 h be5 c\nt$1 he !n holds up "o f+]1 & ! fir/ be+ ready to beg9 aga91 has no f+]s ext5d$1 & ! numb] obta9$ is elev5 t5s1 or t5 t5s & "o t51 or "o hundr$ & t54 ,?is is ! case 9 ,,iii4 ,y w n[ f9d no difficulty ) ! o!r numb]s 9 ! table4 #ab4 ,9 all ^! numb]s a figure 9 ! fir/ column /&s = only z _m y>ds z are writt5 "u t figure 9 7#f74 ,a figure 9 ! second column /&s1 n = z _m y>ds1 b = z _m t5s ( y>ds2 a figure 9 ! ?ird column /&s = z _m hundr$s ( y>ds2 9 ! f\r? column = z _m ?\s&s ( y>ds2 & s on3 t is1 if we suppose a figure to move f any column to ! "o on its left1 x /&s = t5 times z _m y>ds z be=e4 ,recollect ?is1 & y may cease to draw ! l9es betwe5 ! columns1 because ea* figure w be suffici5tly well kn[n by ! place 9 :i* x is2 t is1 by ! numb] ( figures :i* come ^u ! "r h& ( x4 #ac4 ,x is important to recollect t ?is way ( writ+ numb]s1 :i* has become s famili> z to seem ! natural me?od1 is n m natural ?an any o!r4 ,= example1 we might agree to signify "o t5 by ! figure ( "o ) an acc5t1 ?us1 #a2 tw5ty or two t5s by #b2 & s on3 "o hundr$ or t5 t5s by #a2 two hundr$ by #b2 "o ?\s& by #a2 & s on3 putt+ ,roman figures = acc5ts :5 !y become too _m to write ) conv5i5ce4 ,! f\r? numb] 9 ! table w\ld !n be writt5 #b #c #d #h1 :i* might also be express$ by #h #d #c #b1 #d #h #c #b2 or ! ord] ( ! figures might be *ang$ 9 any way1 because _! mean+ dep5ds ^u ! acc5ts :i* are atta*$ to !m1 & n ^u ! place 9 :i* !y /&4 ,h5ce1 a ciph] w\ld nev] be necess>y2 = #ajd w\ld be di/+ui%$ f #ad by writ+ = ! fir/ #a #d1 & = ! second #a #d4 ,! common me?od is pref]r$1 n because x is m exact ?an ?is1 b because x is m simple4 #ad4 ,! di/9ction betwe5 our me?od ( num]ation & t ( ! anci5ts1 is 9 ! mean+ ( ea* figure dep5d+ p>tly ^u ! place 9 :i* x /&s4 ,?us1 9 #ddddd ea* f\r /&s = f\r ( some?+2 b 9 ! fir/ column on ! "r x signifies only f\r ( ! pebbles :i* are c\nt$2 9 ! second1 x means f\r collections ( t5 pebbles ea*2 9 ! ?ird1 f\r ( "o hundr$ ea*2 & s on4 #ae4 ,! ?+s measur$ 9 7#aa7 7 y>ds ( clo?4 ,9 ?is case "o y>d ( clo? is call$ ! unit4 ,! fir/ figure on ! "r is said to be 9 ! units' place1 because x only /&s = s _m units z are 9 ! numb] t is writt5 "u x 9 7#f74 ,! second figure is said to be 9 ! t5s' place1 because x /&s = a numb] ( t5s ( units4 ,! ?ird1 f\r?1 & fif? figures are 9 ! places ( ! hundr$s1 ?\s&s1 & t5s ( ?\s&s1 = a simil> reason4 #af4 ,if ! quantity measur$ _h be5 acres ( l&1 an acre ( l& w\ld h be5 call$ ! unit1 = ! unit is "o ( ! ?+s :i* are measur$4 ,quantities are ( two sorts2 ^? :i* conta9 an exact numb] ( units1 z #dg y>ds1 & ^? :i* d n1 z #dg y>ds & a half4 ,( ^!1 = ! pres5t1 we only consid] ! fir/4 #ag4 ,9 mo/ p>ts ( ari?metic1 all quantities mu/ h ! same unit4 ,y _c say t #b y>ds & #c feet make #e y>ds or #e feet1 because #b & #c make #e2 yet y may say t #b y>ds & #c y>ds make #e y>ds1 & t #b feet & #c feet make #e feet4 ,x w\ld be absurd to try to measure a quantity ( "o k9d ) a unit :i* is a quantity ( ano!r k9d2 = example1 to attempt to tell h[ _m y>ds !re are 9 a gallon1 or h[ _m bu%els ( corn !re are 9 a b>rel ( w9e4 #ah4 ,all ?+s :i* are true ( "s numb]s ( "o unit are true ( ! same numb]s ( any o!r unit4 ,?us1 #ae pebbles & #g pebbles toge!r make #bb pebbles2 #ae acres & #g acres toge!r make #bb acres1 & s on4 ,f ?is we come to say t #ae & #g make #bb1 mean+ t #ae ?+s ( ! same k9d1 & #g m ( ! same k9d z ! fir/1 toge!r make #bb ( t k9d1 :e!r ! k9d m5tion$ be pebbles1 horsem51 acres ( l&1 or any o!r4 ,= ^! x is b necess>y to say1 once = all1 t #ae & #g make #bb4 ,!re=e1 9 future1 on ?is "p ( ! subject ,i %all cease to talk ( any p>ticul> units1 su* z pebbles or acres1 & speak ( numb]s only4 ,a numb]1 consid]$ )\t int5d+ to allude to any p>ticul> ?+s1 is call$ an ab/ract numb]3 & x !n m]ely signifies repetitions ( a unit1 or ! numb] ( times a unit is repeat$4 #ai4 ,i w n[ repeat ! pr9cipal ?+s :i* h be5 m5tion$ 9 ?is *apt]4 ,i4 ,t5 signs are us$1 "o to /& = no?+1 ! re/ = ! fir/ n9e numb]s4 ,!y are #j1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i4 ,! fir/ ( ^! is call$ a ciph]4 ,,ii4 ,high] numb]s h n signs = !mselves1 b are signifi$ by plac+ ! signs already m5tion$ by ! side ( ea* o!r1 & agree+ t ! fir/ figure on ! "r h& %all keep ! value :i* x has :5 x /&s alone2 t ! second on ! "r h& %all mean t5 times z _m z x does :5 x /&s alone2 t ! ?ird figure %all mean "o hundr$ times z _m z x does :5 x /&s alone2 ! f\r?1 "o ?\s& times z _m2 & s on4 ,,iii4 ,! "r h& figure is said to be 9 ! units' place1 ! next to t 9 ! t5s' place1 ! ?ird 9 ! hundr$s' place1 & s on4 ,,iv4 ,:5 a numb] is itself an exact numb] ( t5s1 hundr$s1 or ?\s&s1 c41 z _m ciph]s mu/ be plac$ on ! "r ( x z w br+ ! numb] into ! place :i* is int5d$ = x4 ,! foll[+ are examples3 ,fifty1 or five t5s1 #ej3 sev5 hundr$1 #gjj4 ,five hundr$ & tw5ty-eight ?\s&1 #ebhjjj4 ,if x 7 n = ! ciph]s1 ^! numb]s w\ld be mi/ak5 = #e1 #g1 & #ebh4 ,v4 ,a ciph] 9 ! middle ( a numb] becomes necess>y :5 any "o ( ! d5om9ations1 units1 t5s1 c4 is want+4 ,?us1 tw5ty ?\s& & six is #bjjjf1 two hundr$ & six is #bjf4 ,ciph]s might be plac$ at ! beg9n+ ( a numb]1 b !y w\ld h no mean+4 ,?us #jbf is ! same z #bf1 s9ce ! ciph] m]ely %ews t !re are no hundr$s1 :i* is evid5t f ! numb] itself4 #bj4 ,if we take out ( a numb]1 z #afghe1 any ( ^? figures :i* come toge!r1 z #fg1 & ask1 :at does ?is sixty-sev5 mean8 ( :at is x sixty-sev58 ! answ] is1 sixty-sev5 ( ! same collections z ! #g1 :5 x 0 9 ! numb]2 t is1 #fg hundr$s4 ,= ! #f is #f ?\s&s1 or #f t5 hundr$s1 or sixty hundr$s2 :i*1 ) ! #g1 or #g hundr$s1 is #fg hundr$s3 simil>ly1 ! #fgh is #fgh t5s4 ,?is numb] may !n be express$ ei!r z #a t5 ?\s& #f ?\s&s #g hundr$s #h t5s & #e2 or #af ?\s&s #gh t5s & #e2 or #a t5 ?\s& #fgh t5s & #e2 or #afg hundr$s #h t5s & #e2 or #afgh t5s & #e1 & s on4 #ba4 ,,ex]cises4 ,i4 ,write d[n ! signs =--f\r hundr$ & sev5ty-six2 two ?\s& & n9ety-sev52 sixty-f\r ?\s& ?ree hundr$ & fifty2 two millions sev5 hundr$ & f\r2 five hundr$ & sev5ty-eight millions ( millions4 ,,ii4 ,write at full l5g? #ec1 #ahje1 #ahcj1 #ffgjg1 #ahjiagcbd1 #ffgacgba1 #ijigfcij1 #bejjjjjj4 ,,iii4 ,:at alt]ation takes place 9 a numb] made up entirely ( n9es1 su* z #iiiii1 by add+ "o to x8 ,,iv4 ,%ew t a numb] :i* has five figures 9 x mu/ be great] ?an "o :i* has f\r1 ?\gh ! fir/ h none b small figures 9 x1 & ! second none b l>ge ones4 ,= example1 t #ajaaa is great] ?an #ihgi4 #bb4 ,y n[ see t ! conv5i5ce ( our me?od ( num]ation arises f a few simple signs be+ made to *ange _! value z !y *ange ! column 9 :i* !y are plac$4 ,! same advantage arises f c\nt+ 9 a simil> way all ! articles :i* are us$ 9 e-day life4 ,= example1 we c\nt money by divid+ x into p\nds1 %ill+s1 & p5ce1 ( :i* a %ill+ is #ab p5ce1 & a p\nd #bj %ill+s1 or #bdj p5ce4 ,we write a numb] ( p\nds1 %ill+s1 & p5ce 9 ?ree columns1 g5]ally plac+ po9ts betwe5 ! columns4 ,?us1 #bfc p5ce w\ld n be writt5 z #bfc1 b z #a4 #a4 #aa1 :]e %ews t ! #a 9 ! fir/ column is a p\nd4 ,"h is a sy/em ( num]ation 9 :i* a numb] 9 ! second column on ! "r means #ab times z mu* z ! same numb] 9 ! fir/2 & "o 9 ! ?ird column is tw5ty times z great z ! same 9 ! second1 or #bdj times z great z ! same 9 ! fir/4 ,9 ea* ( ! tables ( measures :i* y w h]eaft] meet )1 y w see a sep>ate sy/em ( num]ation1 b ! me?ods ( calculation = all w be ! same4 #bc4 ,9 ord] to make ! language ( ari?metic %ort]1 "s o!r signs are us$4 ,!y are z foll[3 ,i4 #ae #ch means t #ch is to be add$ to #ae1 & is ! same ?+ z #ec4 ,?is is ! sum ( #ae & #ch1 & is read fifte5 plus ?irty-eight 7plus is ! ,lat9 = m74 ,,ii4 #fd-ab means t #ab is to be tak5 away f #fd1 & is ! same ?+ z #eb4 ,?is is ! diff]5ce ( #fd & #ab1 & is read sixty-f\r m9us twelve 7m9us is ! ,lat9 = less74 ,,iii4 #i #h means t #h is to be tak5 #i times1 & is ! same ?+ z #gb4 ,?is is ! product ( #i & #h1 & is read n9e into eight4 ,,iv4 #ajh_/f means t #ajh is to be divid$ by #f1 or t y mu/ f9d out h[ _m sixes !re are 9 #ajh2 & is ! same ?+ z #ah4 ,?is is ! quoti5t ( #ajh & #f2 & is read a hundr$ & eight by six4 ,v4 ,:5 two numb]s1 or collections ( numb]s1 ) ! =ego+ signs1 are ! same1 ! sign is put betwe5 !m4 ,?us1 t #g & #e make #ab1 is writt5 9 ?is way1 #g #e #ab4 ,?is is call$ an equation1 & is read1 sev5 plus five equals twelve4 ,x is pla9 t we may con/ruct z _m equations z we please4 ,?us3 #ab #g #i - #c #ab #a2 --- - #a #c #b #aa1 #b & s on4 #bd4 ,x (t5 becomes necess>y to speak ( some?+ :i* is true n ( any "o numb] only1 b ( all numb]s4 ,= example1 take #aj & #g2 _! sum#d is #ag1 _! diff]5ce is #c4 ,if ?is sum & diff]5ce be add$ toge!r1 we get #bj1 :i* is twice ! great] ( ! two numb]s fir/ *os54 ,if f #ag we take #c1 we get #ad1 :i* is twice ! less ( ! two numb]s4 ,! same ?+ w be f\nd to hold good ( any two numb]s1 :i* gives ?is g5]al proposition1--,if ! sum & diff]5ce ( two numb]s be add$ toge!r1 ! result is twice ! great] ( ! two2 if ! diff]5ce be tak5 f ! sum1 ! result is twice ! less] ( ! two4 ,if1 !n1 we take any numb]s1 & call !m ! fir/ numb] & ! second numb]1 & let ! fir/ numb] be ! great]2 we h #d ,any little computations :i* occur 9 ! re/ ( ?is section may be made on ! f+]s1 or ) c\nt]s4 7#a/ ,no4 #b;d ,no47 7#a/ ,no4 - #b;d ,no47 twice #a/ ,no4 7#a/ ,no4 #b;d ,no47 - 7#a/ ,no4 - #b;d ,no47 twice #b;d ,no4 ,! brackets "h enclose ! ?+s :i* mu/ be fir/ done1 be=e ! signs :i* jo9 ! brackets are made use (4 ,?us1 #h-7b #a7 7#a #a7 signifies t #b #a mu/ be tak5 #a #a times1 & ! product mu/ be subtract$ f #h4 ,9 ! same mann]1 any result made f two or m numb]s1 :i* is true :atev] numb]s are tak51 may be repres5t$ by us+ fir/ ,no41 second ,no41 c41 to /& = !m1 & by ! signs 9 7#bc74 ,b ?is may be mu* %ort5$2 = z fir/ ,no41 second ,no41 c41 may mean any numb]s1 ! lett]s a & b may be us$ in/ead ( ^! words2 & x mu/ n[ be recollect$ t a & b /& = two numb]s1 provid$ only t a is great] ?an b4 ,let twice a be repres5t$ by #b;a1 & twice b by #b;b4 ,! equations !n become 7a b7 7a - b7 #b;a1 & 7a b7 - 7a - b7 #b;b4 ,?is may be expla9$ /ill fur!r1 z foll[s3 #be4 ,suppose a numb] ( seal$ packets1 m>k$ a1 b1 c1 d1 c41 on ! outside1 ea* ( :i* conta9s a di/9ct b unkn[n numb] ( c\nt]s4 ,z long z we d n "k h[ _m c\nt]s ea* conta9s1 we c make ! lett] :i* belongs to ea* /& = its numb]1 s z to talk ( ! numb] a1 in/ead ( ! numb] 9 ! packet m>k$ a4 ,& because we d n "k ! numb]s1 x does n !re=e foll[ t we "k no?+ :atev] ab\t !m2 = !re are "s connexions :i* exi/ betwe5 all numb]s1 :i* we call g5]al prop]ties ( numb]s4 ,= example1 take any numb]1 multiply x by itself1 & subtract "o f ! result2 & !n subtract "o f ! numb] itself4 ,! fir/ ( ^! w always conta9 ! second exactly z _m times z ! orig9al numb] increas$ by "o4 ,take ! numb] #f2 ?is multipli$ by itself is #cf1 :i* dim9i%$ by "o is #ce2 aga91 #f dim9i%$ by #a is #e2 & #ce conta9s #e1 #g times1 t is1 #f #a times4 ,?is w be f\nd to be true ( any numb]1 &1 :5 prov$1 may be said to be true ( ! numb] conta9$ 9 ! packet m>k$ a1 or ( ! numb] a4 ,if we repres5t a multipli$ by itself by aa1#e we h1 by 7#bc7 aa - #a ------------- a #a4 a - #a #e ,?is %\ld be 7#bc7 a a1 b ! sign is unnecess>y "h4 ,x is us$ ) numb]s1 z 9 #b #g1 to prev5t conf\nd+ ?is1 :i* is #ad1 ) #bg4 #bf4 ,:51 !re=e1 we wi% to talk ( a numb] )\t specify+ any "o 9 p>ticul>1 we use a lett] to repres5t x4 ,?us3 ,suppose we wi% to reason ^u :at w foll[ f divid+ a numb] into ?ree p>ts1 )\t consid]+ :at ! numb] is1 or :at are ! p>ts into :i* x is divid$4 ,let a /& = ! numb]1 & b1 c1 & d1 = ! p>ts into :i* x is divid$4 ,!n1 by our supposition1 a b c d4 ,on ?is we c reason1 & produce results :i* d n belong to any p>ticul> numb]1 b are true ( all4 ,?us1 if "o "p be tak5 away f ! numb]1 ! o!r two w rema91 or a - b c d4 ,if ea* "p be d\bl$1 ! :ole numb] w be d\bl$1 or #b;a #b;b #b;c #b;d4 ,if we dim9i% "o ( ! p>ts1 z d1 by a numb] x1 we dim9i% ! :ole numb] j z mu*1 or a - x b c 7d - x74 #bg4 ,,ex]cises4 ,:at is a #b;b - c1 :]e a #ab1 b #ah1 c #g8--;,answ]1 #da4 aa - bb ,:at is ----------- 1 a - b :]e a #f & b #b8--;,ans4 #h4 ,:at is ! diff]5ce betwe5 7a b77c d7 & a bc d1 = ! foll[+ values ( a1 b1 c1 & d8 a b c d ,ans4 #a #b #c #d #aj #b #ab #g #a #be #a #a #a #a #a ,,section ,,ii4 ,,addition ,,& ,,subtraction4 #bh4 ,!re is no process 9 ari?metic :i* does n consi/ entirely 9 ! increase or dim9ution ( numb]s4 ,!re is !n no?+ :i* might n be done ) collections ( pebbles4 ,probably1 at fir/1 ei!r ^! or ! f+]s 7 us$4 ,our ^w calculation is d]iv$ f ! ,lat9 ^w calculus1 :i* means a pebble4 ,%ort] ways ( c\nt+ h be5 inv5t$1 by :i* _m calculations1 :i* w\ld require long & t$i\s reckon+ if pebbles 7 us$1 are made at once ) v little tr\ble4 ,! f\r great me?ods are1 ,addition1 ,subtraction1 ,multiplication1 & ,division2 ( :i*1 ! la/ two are only ways ( do+ sev]al ( ! fir/ & second at once4 #bi4 ,:5 "o numb] is increas$ by o!rs1 ! numb] :i* is z l>ge z all ! numb]s toge!r is call$ _! sum4 ,! process ( f9d+ ! sum ( two or m numb]s is call$ ,,addition1 &1 z 0 said be=e1 is d5ot$ by plac+ a cross 77 betwe5 ! numb]s :i* are to be add$ toge!r4 ,suppose x requir$ to f9d ! sum ( #ahcd & #bgii4 ,9 ord] to add ^! numb]s1 take !m to pieces1 divid+ ea* into its units1 t5s1 hundr$s1 & ?\s&s3 #ahcd is #a ?\s4 #h hund4 #c t5s & #d2 #bgii is #b ?\s4 #g hund4 #i t5s & #i4 ,ea* numb] is ?us brok5 up into f\r p>ts4 ,if to ea* "p ( ! fir/ y add ! "p ( ! second :i* is "u x1 & !n put toge!r :at y get f ^! additions1 y w h add$ #ahcd & #bgii4 ,9 ! fir/ numb] are #d units1 & 9 ! second #i3 ^! w1 :5 ! numb]s are add$ toge!r1 contribute #ac units to ! sum4 ,aga91 ! #c t5s 9 ! fir/ & ! #i t5s 9 ! second w contribute #ab t5s to ! sum4 ,! #h hundr$s 9 ! fir/ & ! #g hundr$s 9 ! second w add #ae hundr$s to ! sum2 & ! ?\s& 9 ! fir/ ) ! #b ?\s&s 9 ! second w contribute #c ?\s&s to ! sum2 !re=e ! sum requir$ is #c ?\s&s1 #ae hundr$s1 #ab t5s1 & #ac units4 ,to simplify ?is result1 y mu/ recollect t-- #ac units are #a t5 & #c units4 #ab t5s are #a hund4 & #b t5s4 #ae hund4 are #a ?\s4 & #e hund4 #c ?\s4 are #c ?\s4 ,n[ collect ! numb]s on ! "r h& side toge!r1 z 0 done be=e1 & ?is w give1 z ! sum ( #ahcd & #bgii1 #d ?\s&s1 #f hundr$s1 #c t5s1 & #c units1 :i* 7#ai7 is writt5 #dfcc4 #cj4 ,! =m] process1 writt5 ) ! signs ( 7#bc7 is z foll[s3 #ahcd #a #ajjj #h #ajj #c #aj #d #bgii #b #ajjj #g #ajj #i #aj #i ,!re=e1 #ahcd #bgii #c #ajjj #ae #ajj #ab #aj #ac ,b #ac #a #aj #c #ab #aj #a #ajj #b #aj #ae #ajj #a #ajjj #e #ajj #c #ajjj #c #ajjj ,!re=e1 #ahcd #bgii #d #ajjj #f #ajj #c #aj #c #dfcc4 #ca4 ,! same process is to be foll[$ 9 all cases1 b n at ! same l5g?4 ,9 ord] to be able to g ?r\gh x1 y mu/ "k h[ to add toge!r ! simple numb]s4 ,?is c only be done by memory2 & to help ! memory y %\ld make ! foll[+ table ?ree or f\r times = y\rself3 ---------------------------------------- #a #b #c #d #e #f #g #h #i ---------------------------------------- #a #b #c #d #e #f #g #h #i #aj ---------------------------------------- #b #c #d #e #f #g #h #i #aj #aa ---------------------------------------- #c #d #e #f #g #h #i #aj #aa #ab ---------------------------------------- #d #e #f #g #h #i #aj #aa #ab #ac ---------------------------------------- #e #f #g #h #i #aj #aa #ab #ac #ad ---------------------------------------- #f #g #h #i #aj #aa #ab #ac #ad #ae ---------------------------------------- #g #h #i #aj #aa #ab #ac #ad #ae #af ---------------------------------------- #h #i #aj #aa #ab #ac #ad #ae #af #ag ---------------------------------------- #i #aj #aa #ab #ac #ad #ae #af #ag #ah ---------------------------------------- ,! use ( ?is table is z foll[s3 ,suppose y want to f9d ! sum ( #h & #g4 ,look 9 ! left-h& column = ei!r ( !m1 #h1 = example2 & look 9 ! top column = #g4 ,on ! same l9e z #h1 & und]nea? #g1 y f9d #ae1 _! sum4 #cb4 ,:5 ?is table has be5 ?or\ghly committ$ to memory1 s t y c tell at once ! sum ( any two numb]s1 nei!r ( :i* exce$s #i1 y %\ld ex]cise y\rself 9 add+ & subtract+ two numb]s1 "o ( :i* is great] ?an #i & ! o!r less4 ,y %\ld write d[n a great numb] ( su* s5t5ces z ! foll[+1 :i* w ex]cise y at ! same "t 9 addition1 & 9 ! use ( ! signs m5tion$ 9 7#bc74 #ab #f #ah #bb #f #bh #ai #h #bg #ed #i #fc #ef #g #fc #bb #h #cj #ajj - #i #ia #bg - #h #ai #dd - #f #ch1 c4 #cc4 ,:5 ! la/ two articles h be5 ?or\ghly /udi$1 y w be able to f9d ! sum ( any numb]s by ! foll[+ process1#f :i* is ! same z t 9 7#bi74 #f ,9 ?is & all o!r processes1 ! /ud5t is /rongly recomm5d$ to look at & foll[ ! fir/ ,app5dix4 ,,rule ,i4 ,place ! numb]s "u "o ano!r1 units "u units1 t5s "u t5s1 & s on4 ,,ii4 ,add toge!r ! units ( all1 & "p ! :ole numb] ?us obta9$ into units & t5s4 ,?us1 if #he be ! numb]1 "p x into #h t5s & #e units2 if #acf be ! numb]1 "p x into #ac t5s & #f units 7#bj74 ,,iii4 ,write d[n ! units ( ?is numb] "u ! units ( ! re/1 & keep 9 memory ! numb] ( t5s4 ,,iv4 ,add toge!r all ! numb]s 9 ! column ( t5s1 rememb]+ to take 9 7or c>ry1 z x is call$7 ! t5s :i* y 7 told to recollect 9 ,,iii41 & divide ?is numb] ( t5s into t5s & hundr$s4 ,?us1 if #cce t5s be ! numb] obta9$1 "p ?is into #cc hundr$s & #e t5s4 ,v4 ,place ! numb] ( t5s "u ! t5s1 & rememb] ! numb] ( hundr$s4 ,,vi4 ,proce$ 9 ?is way ?r\gh e column1 & at ! la/ column1 in/ead ( sep>at+ ! numb] y obta9 into two p>ts1 write x all d[n be=e ! re/4 ,,example4--,:at is #ahje #cf #aigbg #c #adgd #bjjh #ahje #cf #aigbg #c #adgd #bjjh ----- #bejec ,! addition ( ! units' l9e1 or #h #d #c #g #f #e1 gives #cc1 t is1 #c t5s & #c units4 ,put #c 9 ! units' place1 & add toge!r ! l9e ( t5s1 tak+ 9 at ! beg9n+ ! #c t5s :i* 7 creat$ by ! addition ( ! units' l9e4 ,t is1 f9d #c #j #g #b #c #j1 :i* gives #ae = ! numb] ( t5s2 t is1 #a hundr$ & #e t5s4 ,add ! l9e ( hundr$s toge!r1 tak+ c>e to add ! #a hundr$ :i* arose 9 ! addition ( ! l9e ( t5s2 t is1 f9d #a #j #d #g #h1 :i* gives exactly #bj hundr$s1 or #b ?\s&s & no hundr$s4 ,put a ciph] 9 ! hundr$s' place 7because1 if y d n1 ! next figure w be tak5 = hundr$s in/ead ( ?\s&s71 & add ! figures 9 ! ?\s&s' l9e toge!r1 rememb]+ ! #b ?\s&s :i* arose f ! hundr$s' l9e2 t is1 f9d #b #b #a #i #a1 :i* gives #ae ?\s&s1 or #a t5 ?\s& & #e ?\s&4 ,write #e "u ! l9e ( ?\s&s1 & collect ! figures 9 ! l9e ( t5s ( ?\s&s1 rememb]+ ! t5 ?\s& :i* arose out ( ! ?\s&s' l9e2 t is1 f9d #a #a1 or #b t5 ?\s&s4 ,write #b "u ! t5 ?\s&s' l9e1 & ! op]ation is complet$4 #cd4 ,z an ex]cise 9 addition1 y may satisfy y\rself t :at ,i n[ say ( ! foll[+ squ>e is correct4 ,! numb]s 9 e r[1 :e!r reckon$ upright1 or f "r to left1 or f corn] to corn]1 :5 add$ toge!r give ! numb] #bdaef4 ---------------------------------------- ---- #bjafdbabafefchebabifcdib #icfcacb #egfbggb #baf ---------------------------------------- ---- #bebbjebdbdhafibchhhaccbcebh #igbcafh #fabbdab ---------------------------------------- ---- #bddh #bhhbjhhdbhdagbhcibdacfhcefdajjhbhjh #fdh ---------------------------------------- ---- #fhdbdhd #cbdbabddcbjagfdcifjadjdcbjdajddbhdd ---------------------------------------- ---- #bhhj #gbjbebj #cfjbafjdcefahjjcfjjaddjcbdjajhj ---------------------------------------- ---- #aaafbiaf #gefbeef #cifbaifciifahcfcfcfadgfcbgf ---------------------------------------- ---- #ccabaaebbieb #gibbeib #cfbbcbdjcbahgbcfgbaeab ---------------------------------------- ---- #aedhccdhaahhbihh #dcbbfbh #gbbbfhdjfhaijhcgjh ---------------------------------------- ---- #cgddaehdcchd #hbhcjbd #dfhbffd #ajhbcjddajdaidd ---------------------------------------- ---- #aihjcghjabbdcdbj #hfdcjfj #ejdbgjj #addbcdjdadj ---------------------------------------- ---- #dagfafbjchafabfjcdef #ijjcjif #edjbgcf #ahjbcgf ---------------------------------------- ---- #ce4 ,if two numb]s mu/ be add$ toge!r1 x w n alt] ! sum if y take away a "p ( "o1 provid$ y put on z mu* to ! o!r4 ,x is pla9 t y w n alt] ! :ole numb] ( a collection ( pebbles 9 two baskets by tak+ any numb] out ( "o1 & putt+ !m into ! o!r4 ,?us1 #ae #g is ! same z #ab #aj1 s9ce #ab is #c less ?an #ae1 & #aj is ?ree m ?an #g4 ,?is 0 ! pr9ciple ^u :i* ! :ole ( ! process 9 7#bi7 0 conduct$4 #cf4 ,let a & b /& = two numb]s1 z 9 7#bd74 ,x is impossible to tell :at _! sum w be until ! numb]s !mselves are kn[n4 ,9 ! mean :ile a b /&s = ?is sum4 ,to say1 9 algebraical language1 t ! sum ( a & b is n alt]$ by add+ c to a1 provid$ we take away c f b1 we h ! foll[+ equation3 7a c7 7b - c7 a b2 :i* may be writt5 )\t brackets1 ?us1 a c b - c a b4 ,= ! mean+ ( ^! two equations w appe> to be ! same1 on consid]ation4 #cg4 ,if a be tak5 twice1 ?ree times1 c41 ! results are repres5t$ 9 algebra by #b;a1 #c;a1 #d;a1 c4 ,! sum ( any two ( ?is s]ies may be express$ 9 a %ort] =m ?an by writ+ ! sign betwe5 !m2 = ?\gh we d n "k :at numb] a /&s =1 we "k t1 be x :at x may1 #b;a #b;a #d;a1 #c;a #b;a #e;a1 #d;a #i;a #ac;a2 & g5]ally1 if a tak5 m times be add$ to a tak5 n times1 ! result is a tak5 m n times1 or ma na 7m n7a4 #ch4 ,! use ( ! brackets mu/ "h be notic$4 ,!y mean1 t ! expression conta9$ inside !m mu/ be us$ exactly z a s+le lett] w\ld be us$ 9 ! same place4 ,?us1 pa signifies t a is tak5 p times1 & 7m n7a1 t a is tak5 m n times4 ,x is1 !re=e1 a diff]5t ?+ f m na1 :i* means t a1 aft] be+ tak5 n times1 is add$ to m4 ,?us 7#c #d7 #b is #g #b or #ad2 :ile #c #d #b is #c #h1 or #aa4 #ci4 ,:5 "o numb] is tak5 away f ano!r1 ! numb] :i* is left is call$ ! diff]5ce or rema9d]4 ,! process ( f9d+ ! diff]5ce is call$ ,,subtraction4 ,! numb] :i* is to be tak5 away mu/ be ( c\rse ! less] ( ! two4 #dj4 ,! process ( subtraction dep5ds ^u ^! two pr9ciples4 ,i4 ,! diff]5ce ( two numb]s is n alt]$ by add+ a numb] to ! fir/1 if y add ! same numb] to ! second2 or by subtract+ a numb] f ! fir/1 if y subtract ! same numb] f ! second4 ,conceive two baskets ) pebbles 9 !m1 9 ! fir/ ( :i* are #ajj pebbles m ?an 9 ! second4 ,if ,i put #ej m pebbles into ea* ( !m1 !re are /ill only #ajj m 9 ! fir/ ?an 9 ! second1 & ! same if ,i take #ej f ea*4 ,!re=e1 9 f9d+ ! diff]5ce ( two numb]s1 if x %\ld be conv5i5t1 ,i may add any numb] ,i please to bo? ( !m1 because1 ?\gh ,i alt] ! numb]s !mselves by s do+1 ,i d n alt] _! diff]5ce4 ,,ii4 ,s9ce #f exce$s #d by #b1 & #c exce$s #b by #a1 & #ab exce$s #e by #g1 #f1 #c1 & #ab toge!r1 or #ba1 exce$ #d1 #b1 & #e toge!r1 or #aa1 by #b1 #a1 & #g toge!r1 or #aj3 ! same ?+ may be said ( any o!r numb]s4 #da4 ,if a1 b1 & c be ?ree numb]s1 ( :i* a is great] ?an b 7#dj71 ,i4 leads to ! foll[+1 7a c7 - 7b c7 a - b4 ,aga91 if c be less ?an a & b1 7a - c7 - 7b - c7 a - b4 ,! brackets _c be "h remov$ z 9 7#cf74 ,t is1 p- 7q-r7 is n ! same ?+ z p-q- r4 ,=1 9 ! fir/1 ! diff]5ce ( q & r is subtract$ f p2 b 9 ! second1 fir/ q & !n r are subtract$ f p1 :i* is ! same z subtract+ z mu* z q & r toge!r1 or q r4 ,!re=e p-q-r is p-7q r74 ,9 ord] to %ew h[ to remove ! brackets f p -7q-r7 )\t alt]+ ! value ( ! result1 let u take ! simple in/ance #ab-7h-e74 ,if we subtract #h f #ab1 or =m #ab-h1 we subtract too mu*2 because x is n #h :i* is to be tak5 away1 b z mu* ( #h z is left aft] dim9i%+ x by #e4 ,9 =m+ #ab-h we h !re=e subtract$ #e too mu*4 ,?is mu/ be set "r by add+ #e to ! result1 :i* gives #ab-h #e = ! value ( #ab-7h-e74 ,! same reason+ applies to e case1 & we h !re=e1 p - 7q r7 p - q - r4 p - 7q - r7 p - q r4 ,by ! same k9d ( reason+1 a - 7b c - d - e7 a - b - c d e4 #b;a #c;b - 7a - #b;b7 #b;a #c;b - a #b;b a #e;b4 #dx y - 7#agx - #iy7 #dx y - #agx #iy #ajy - #acx4 #db4 ,i want to f9d ! diff]5ce ( ! numb]s #eggfb & #cdfca4 ,take ^! to pieces z 9 7#bi7 & #eggfb is #e t5-?4 #g ?4 #g hund4 #f t5s & #b units4 #cdfca is #c t5-?4 #d ?4 #f hund4 #c t5s & #a unit4 ,n[ #b units exce$ #a unit by #a unit4 #f t5s #c t5s #c t5s4 #g hundr$s #f hundr$s #a hundr$4 #g ?\s&s #d ?\s&s #c ?\s&s4 #e t5-?\s&s #c t5-?\s4 #b t5-?\s4 ,!re=e1 by 7#dj1 ,pr9ciple ,,ii47 all ! fir/ column toge!r exce$s all ! second column by all ! ?ird column1 t is1 by #b t5-?4 #c ?4 #a hund4 #c t5s & #a unit1 :i* is #bcaca4 ,!re=e ! diff]5ce ( #eggfb & #cdfca is #bcaca1 or #eggfb-cdfca #bcaca4 #dc4 ,suppose ,i want to f9d ! diff]5ce betwe5 #fabgd & #cifbh4 ,write !m at l5g?1 & #fabgd is #f t5-?4 #a ?4 #b hund4 #g t5s & #d units4 #cifbh is #c t5-?4 #i ?4 #f hund4 #b t5s & #h units4 ,if we attempt to d ! same z 9 ! la/ article1 !re is a difficulty imm$iately1 s9ce #h1 be+ great] ?an #d1 _c be tak5 f x4 ,b f 7#dj7 x appe>s t we %all n alt] ! diff]5ce ( two numb]s if we add ! same numb] to bo? ( !m4 ,add t5 to ! fir/ numb]1 t is1 let !re be #ad units in/ead ( f\r1 & add t5 also to ! second numb]1 b in/ead ( add+ t5 to ! numb] ( units1 add "o to ! numb] ( t5s1 :i* is ! same ?+4 ,! numb]s w !n /& ?us1 #f t5-?\s4 #a ?\s4 #b hund4 #g t5s & #ad units4#g #c t5-?\s4 #i ?\s4 #f hund4 #c t5s & #h units4 #g ,^? numb]s :i* h be5 alt]$ are put 9 italics4 ,y n[ see t ! units & t5s 9 ! l[] c be subtract$ f ^? 9 ! upp] l9e1 b t ! hundr$s _c4 ,to rem$y ?is1 add "o ?\s& or #aj hundr$ to bo? numb]s1 :i* w n alt] _! diff]5ce1 & rememb] to increase ! hundr$s 9 ! upp] l9e by #aj1 & ! ?\s&s 9 ! l[] l9e by #a1 :i* are ! same ?+s4 ,& s9ce ! ?\s&s 9 ! l[] _c be subtract$ f ! ?\s&s 9 ! upp] l9e1 add #a t5 ?\s& or #aj ?\s& to bo? numb]s1 & increase ! ?\s&s 9 ! upp] l9e by #aj1 & ! t5 ?\s&s 9 ! l[] l9e by #a1 :i* are ! same ?+s2 & at ! close ! numb]s :i* we get w be1 #f t5-?\s4 #aa ?\s4 #ab hund4 #g t5s & #ad units4 #d t5-?\s4 #aj ?\s4 #f hund4 #c t5s & #h units4 ,^! numb]s are n1 x is true1 ! same z ^? giv5 at ! beg9n+ ( ?is article1 b _! diff]5ce is ! same1 by 7#dj74 ,) ! la/-m5tion$ numb]s proce$ 9 ! same way z 9 7#db71 :i* w give1 z _! diff]5ce1 #b t5-?\s4 #a ?\s4 #f hund4 #d t5s1 & #f units1 :i* is #bafdf4 #dd4 ,f ?is we d$uce ! foll[+ rules = subtraction3 ,i4 ,write ! numb] :i* is to be subtract$ 7:i* is1 ( c\rse1 ! less] ( ! two1 & is call$ ! subtrah5d7 "u ! o!r1 s t its units %all fall "u ! units ( ! o!r1 & s on4 ,,ii4 ,subtract ea* figure ( ! l[] l9e f ! "o above x1 if t c be done4 ,:]e t _c be done1 add t5 to ! upp] figure1 & !n subtract ! l[] figure2 b recollect 9 ?is case always to increase ! next figure 9 ! l[] l9e by #a1 be=e y beg9 to subtract x f ! upp] "o4 #de4 ,if !re %\ld n be z _m figures 9 ! l[] l9e z 9 ! upp] "o1 proce$ z if !re 7 z _m ciph]s at ! beg9n+ ( ! l[] l9e z w make ! numb] ( figures equal4 ,y d n alt] a numb] by plac+ ciph]s at ! beg9n+ ( x4 ,= example1 #jjhah is ! same numb] z #hah1 = x means #j t5-?\s4 #j ?\s4 #h hunds4 #a t5 & #h units2 ! fir/ two signs are no?+1 & ! re/ is #h hundr$s1 #a t51 & #h units1 or #hah4 ,! second does n diff] f ! fir/1 except 9 its be+ said t !re are no ?\s&s & no t5s ( ?\s&s 9 ! numb]1 :i* may be kn[n )\t _! be+ m5tion$ at all4 ,y may ask1 p]haps1 :y ?is does n apply to a ciph] plac$ 9 ! middle ( a numb]1 or at ! "r ( x1 z1 = example1 9 #bhjjg & #cigjj8 ,b y mu/ recollect1 t if x 7 n = ! two ciph]s 9 ! fir/1 ! #h w\ld be tak5 = #h t5s1 in/ead ( #h ?\s&s2 & if x 7 n = ! ciph]s 9 ! second1 ! #g w\ld be tak5 = #g units1 in/ead ( #g hundr$s4 #df4 ,,example4 ,:at is ! diff]5ce betwe5 #cgjhbiafdjjcjagd & #cjhacfdibgfahh ---------------- ,diff]5ce #cfggdggiijgecihf ,,ex]cises4 ,i4 ,:at is #ahccg #adibfcbjj - #fdgbijb8--;,answ] #adbhjhfce4 ,:at is #ajjj - #dfd #cbgi - #fdf8--;,ans4 #cafi4 ,,ii4 ,subtract #fd #gf #add - #ah f #cc - #b #ajjjcg4--;,ans4 #iihjb4 ,,iii4 ,:at %ort] rule might be made = subtraction :5 all ! figures 9 ! upp] l9e are ciph]s except ! fir/8 = example1 9 f9d+ #ajjjjjjj - #bgcafcd4 ,,iv4 ,f9d #ahcfb #bdfi & #ahcfb-bdfi1 add ! second result to ! fir/1 & !n subtract #ahcfb2 subtract ! second f ! fir/1 & !n subtract #bdfi4--;,answ] #ahcfb & #bdfi4 ,v4 ,!re are f\r places on ! same l9e 9 ! ord] ,a1 ,b1 ,c1 & ,d4 ,f ,a to ,d x is #adfc miles2 f ,a to ,c x is #gbh miles2 & f ,b to ,d x is #acag miles4 ,h[ f> is x f ,a to ,b1 f ,b to ,c1 & f ,c to ,d8--,answ]4 ,f ,a to ,b #adf1 f ,b to ,c #ehb1 & f ,c to ,d #gce miles4 ,,vi4 ,9 ! foll[+ table subtract ,b f ,a1 & ,b f ! rema9d]1 & s on until ,b c be no long] subtract$4 ,f9d h[ _m times ,b c be subtract$ f ,a1 & :at is ! la/ rema9d]4 ,no4 ( ,a ,b times4 ,rema9d]4 #bcfjd #iiii #b #cfjf #bjiifa #cgagc #e #bdjif #gdgab #fgib #aa #j #dhjbdfi #fedcba #g #bbbbbb #ahhdigdg #cadaeib #f #aie #ihgfedcba #abcdefghi #h #i ,,section ,,iii4 ,,multiplication4 #dg4 ,i h said t all que/ions 9 ari?metic require no?+ b addition & subtraction4 ,i d n mean by ?is t no rule %\ld ev] be us$ except ^? giv5 9 ! la/ section1 b t all o!r rules only %ew %ort] ways ( f9d+ :at might be f\nd1 if we pleas$1 by ! me?ods !re d$uc$4 ,ev5 ! la/ two rules !mselves are only %ort & conv5i5t ways ( do+ :at may be done ) a numb] ( pebbles or c\nt]s4 #dh4 ,i want to "k ! sum ( five sev5te5s1 or ,i ask ! foll[+ "q3 ,!re are five heaps ( pebbles1 & sev5te5 pebbles 9 ea* heap2 h[ _m are !re 9 all8 ,write five sev5te5s 9 a column1 & make ! addition1 :i* gives #he4 ,9 ?is case #he is call$ ! product ( #e & #ag1 & ! process ( f9d+ ! product is call$ ,,multiplication1 :i* gives no?+ m ?an ! addition ( a numb] ( ! same quantities4 ,"h #ag is call$ ! multiplic&1 & #e is call$ ! multipli]4 #ag #ag #ag #ag #ag ---- #he #di4 ,if no "q h>d] ?an ?is 7 ev] propos$1 !re w\ld be no occasion = a %ort] way ?an ! "o "h foll[$4 ,b if !re 7 #acfg heaps ( pebbles1 & #dbi 9 ea* heap1 ! :ole numb] is !n #acfg times #dbi1 or #dbi multipli$ by #acfg4 ,i %\ld h to write #dbi #acfg times1 & !n to make an addition ( enorm\s l5g?4 ,to avoid ?is1 a %ort] rule is necess>y1 :i* ,i n[ proce$ to expla94 #ej4 ,! /ud5t mu/ fir/ make himself acqua9t$ ) ! products ( all numb]s z f> z #aj times #aj by means ( ! foll[+ table1#h :i* mu/ be committ$ to memory4 ---------------------------------------- -------- #a #b #c #d #e #f #g #h #i #aj #aa #ab ---------------------------------------- -------- #b #d #f #h #aj #ab #ad #af #ah #bj #bb #bd ---------------------------------------- -------- #c #f #i #ab #ae #ah #ba #bd #bg #cj #cc #cf ---------------------------------------- -------- #d #h #ab #af #bj #bd #bh #cb #cf #dj #dd #dh ---------------------------------------- -------- #e #aj #ae #bj #be #cj #ce #dj #de #ej #ee #fj ---------------------------------------- -------- #f #ab #ah #bd #cj #cf #db #dh #ed #fj #ff #gb ---------------------------------------- -------- #g #ad #ba #bh #ce #db #di #ef #fc #gj #gg #hd ---------------------------------------- -------- #h #af #bd #cb #dj #dh #ef #fd #gb #hj #hh #if ---------------------------------------- -------- #i #ah #bg #cf #de #ed #fc #gb #ha #ij #ii #ajh ---------------------------------------- -------- #aj #bj #cj #dj #ej #fj #gj #hj #ij #ajj #aaj #abj ---------------------------------------- -------- #aa #bb #cc #dd #ee #ff #gg #hh #ii #aaj #aba #acb ---------------------------------------- -------- #ab #bd #cf #dh #fj #gb #hd #if #ajh #abj #acb #add ---------------------------------------- -------- #h ,z x is usual to le>n ! product ( numb]s up to #ab times #ab1 ,i h ext5d$ ! table ?us f>4 ,9 my op9ion1 all pupils :o %ew a tol]able capacity %\ld sl[ly commit ! products to memory z f> z #bj times #bj1 9 ! c\rse ( _! progress ?r\gh ?is "w4 ,if f ?is table y wi% to "k :at is #g times #f1 look 9 ! fir/ upright column on ! left = ei!r ( !m2 #f = example4 ,proce$ to ! "r until y come into ! column m>k$ #g at ! top4 ,y !re f9d #db1 :i* is ! product ( #f & #g4 #ea4 ,y may f9d1 9 ?is way1 ei!r #f times #g1 or #g times #f1 & = bo? y f9d #db4 ,t is1 six sev5s is ! same numb] z sev5 sixes4 ,?is may be %ewn z foll[s3 ,place sev5 c\nt]s 9 a l9e1 & repeat t l9e 9 all six times4 ,! numb] ( c\nt]s 9 ! :ole is #f times #g1 or six sev5s1 if ,i reckon ! r[s f ! top to ! bottom2 b if ,i c\nt ! r[s t /& side by side1 ,i f9d sev5 ( !m1 & six 9 ea* r[1 ! :ole numb] ( :i* is #g times #f1 or sev5 sixes4 ,& ! :ole numb] is #db1 :i*ev] way ,i c\nt4 ,! same me?od may be appli$ to any o!r two numb]s4 ,if ! signs ( 7#bc7 7 us$1 x w\ld be said t #g #f #f #g4 #eb4 ,to take any quantity a numb] ( times1 x w be en\gh to take e "o ( its p>ts ! same numb] ( times4 ,?us1 a sack ( corn w be increas$ fifty-fold1 if ea* bu%el :i* x conta9s be replac$ by #ej bu%els4 ,a c\ntry w be d\bl$ by d\bl+ e acre ( l&1 or e c\nty1 :i* x conta9s4 ,simple z ?is may appe>1 x is necess>y to /ate x1 because x is "o ( ! pr9ciples on :i* ! rule ( multiplication dep5ds4 #ec4 ,9 ord] to multiply by any numb]1 y may multiply sep>ately by any p>ts into :i* y *oose to divide t numb]1 & add ! results4 ,= example1 #d & #b make #f4 ,to multiply #g by #f fir/ multiply #g by #d1 & !n by #b1 & add ! products4 ,?is w give #db1 :i* is ! product ( #g & #f4 ,aga91 s9ce #eg is made up ( #cb & #be1 #eg times #ej is made up ( #cb times #ej & #be times #ej1 & s on4 ,if ! signs 7 us$1 ^! w\ld be writt5 ?us3 #g #f #g #d #g #b4 #ej #eg #ej #cb #ej #be4 #ed4 ,! pr9ciples 9 ! la/ two articles may be express$ ?us3 ,if a be made up ( ! p>ts x1 y1 & x1 ma is made up ( mx1 my1 & mz2 or1 if a x y z4 ma mx my mz1 or1 m7x y z7 mx my mz4 ,a simil> result may be obta9$ if a1 in/ead ( be+ made up ( x1 y1 & z1 is made by comb9$ additions & subtractions1 su* z x y-z1 x- y z1 x-y-z1 c4 ,to take ! fir/ z an in/ance3 ,let a x y - z1 !n ma mx my - mz4 ,=1 if a _h be5 x y1 ma w\ld h be5 mx my4 ,b s9ce a is less ?an x y by z1 too mu* by z has be5 repeat$ e "t t x y has be5 repeat$2--t is1 mz too mu* has be5 tak52 consequ5tly1 ma is n mx my1 b mx my-mz4 ,simil> reason+ may be appli$ to o!r cases1 & ! foll[+ results may be obta9$3 m7a b c - d7 ma mb mc - md4 a7a - b7 aa - ab4 b7a - b7 ba - bb4 #c7b;a - #d;b7 #f;a - #ab;b4 #g;a7#g #b;b7 #di;a #ad;ab4 7aa a #a7;a aaa aa a4 7#c;ab - #b;c7#d;abc #ab;aabbc - #h;abcc4 #ee4 ,!re is ano!r way 9 :i* two numb]s may be multipli$ toge!r4 ,s9ce #h is #d times #b1 #g times #h may be made by multiply+ #g & #d1 & !n multiply+ t product by #b4 ,to %ew ?is1 place #g c\nt]s 9 a l9e1 & repeat t l9e 9 all #h times1 z 9 figures ,i4 & ,,ii4 ,i4 --------------- ,a --------------- --------------- ,b --------------- ,,ii4 --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ,! numb] ( c\nt]s 9 all is #h times #g1 or #ef4 ,b 7z 9 fig4 ,i47 enclose ea* f\r r[s 9 oblong figures1 su* z ,a & ,b4 ,! numb] 9 ea* oblong is #d times #g1 or #bh1 & !re are two ( ^? oblongs2 s t 9 ! :ole ! numb] ( c\nt]s is twice #bh1 or #bh x #b1 or #g fir/ multipli$ by #d1 & t product multipli$ by #b4 ,9 figure ,,ii4 x is %ewn t #g multipli$ by #h is also #g fir/ multipli$ by #b1 & t product multipli$ by #d4 ,! same me?od may be appli$ to o!r numb]s4 ,?us1 s9ce #hj is #h times #aj1 #bef times #hj is #bef multipli$ by #h1 & t product multipli$ by #aj4 ,if we use ! signs1 ! =ego+ ass]tions are made ?us3 #g #h #g #d #b #g #b #d4 #bef #hj #bef #h #aj #bef #aj #h4 ,,ex]cises4 ,%ew t #b #c #d #e #b #d #c #e #e #d #b #c1 c4 ,%ew t #ah #ajj #ah #eg #ah #dc4 #ef4 ,articles 7#ea7 & 7#ee7 may be express$ 9 ! foll[+ way1 :]e by ab we mean a tak5 b times2 by abc1 a tak5 b times1 & ! result tak5 c times4 ab ba4 abc acb bca bac1 c4 abc a 7bc7 b 7ca7 c 7ab74 ,if we w\ld say t ! same results are produc$ by multiply+ by b1 c1 & d1 "o aft] ! o!r1 & by ! product bcd at once1 we write ! foll[+3 a b c d a bcd4 ,! fact is1 t if any numb]s are to be multipli$ toge!r1 ! product ( any two or m may be =m$1 & sub/itut$ in/ead ( ^? two or m2 ?us1 ! product abcdef may be =m$ by multiply+ ab cde f abf de c abc def c4 #eg4 ,9 ord] to multiply by #aj1 annex a ciph] to ! "r h& ( ! multiplic&4 ,?us1 #aj times #bcef is #bcefj4 ,to %ew ?is1 write #bcef at l5g? :i* is #b ?\s&s1 #c hundr$s1 #e t5s1 & #f units4 ,take ea* ( ^! p>ts t5 times1 :i*1 by 7#eb71 is ! same z multiply+ ! :ole numb] by #aj1 & x w !n become #b t5s ( ?\4 #c t5s ( hun4 #e t5s ( t5s1 & #f t5s1 :i* is #b t5-?\4 #c ?\s4 #e hun4 & #f t5s4 ,?is mu/ be writt5 #bcefj1 because #f is n to be #f units1 b #f t5s4 ,!re=e #bcef #aj #bcefj4 ,9 ! same way y may %ew1 t 9 ord] to multiply by #ajj y mu/ affix two ciph]s to ! "r2 to multiply by #ajjj y mu/ affix ?ree ciph]s1 & s on4 ,! rule w be be/ caught f ! foll[+ table3 #ac #aj #acj #ac #ajj #acjj #ac #ajjj #acjjj #ac #ajjjj #acjjjj #adb #ajjj #adbjjj #bcgjj #aj #bcgjjj #cjdj #ajjj #cjdjjjj #ajjjj #ajjjjj #ajjjjjjjjj #eh4 ,i n[ %ew h[ to multiply by "o ( ! numb]s1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 or #i4 ,i d n include #a1 because multiply+ by #a1 or tak+ ! numb] once1 is :at is meant by simply writ+ d[n ! numb]4 ,i want to multiply #acfh by #h4 ,write ! fir/ numb] at full l5g?1 :i* is #a ?\s&1 #c hundr$s1 #f t5s1 & #h units4 ,to multiply ?is by #h1 multiply ea* ( ^! p>ts by #h 7#ej7 & 7#eb71 :i* w give #h ?\s&s1 #bd hundr$s1 #dh t5s1 & #fd units4 ,n[ #fd units are writt5 ?us #fd #dh t5s #dhj #bd hundr$s #bdjj #h ?\s&s #hjjj ,add ^! toge!r1 :i* gives #ajidd z ! product ( #acfh & #h1 or #acfh #h #ajidd4 ,by work+ a few examples 9 ?is way y w see = foll[+ rule4 #ei4 ,i4 ,multiply ! fir/ figure ( ! multiplic& by ! multipli]1 write d[n ! units' figure1 & res]ve ! t5s4 ,,ii4 ,d ! same ) ! second figure ( ! multiplic&1 & add to ! product ! numb] ( t5s f ! fir/2 put d[n ! units' figure ( ?is1 & res]ve ! t5s4 ,,iii4 ,proce$ 9 ?is way till y come to ! la/ figure1 & !n write d[n ! :ole numb] obta9$ f t figure4 ,,iv4 ,if !re be a ciph] 9 ! multiplic&1 treat x z if x 7 a numb]1 obs]v+ t #j #a #j1 #j #b #j1 c4 #fj4 ,9 a simil> way a numb] c be multipli$ by a figure :i* is accompani$ by ciph]s1 z1 = example1 #hjjj4 ,= #hjjj is #h #ajjj1 & !re=e 7#ee7 y mu/ fir/ multiply by #h & !n by #ajjj1 :i* la/ op]ation 7#eg7 is done by plac+ #c ciph]s on ! "r4 ,h5ce ! rule 9 ?is case is1 multiply by ! simple numb]1 & place ! numb] ( ciph]s :i* foll[ x at ! "r ( ! product4 ,,example4 ,multiply #afgidbchjjhgb by #fjjjj ------------------ #ajjgfedbhjebcbjjjj #fa4 ,,ex]cises4 ,:at is #ajjgcfj #g8 ,answ]1 #gjeaebj4 #abcdefghi #i #aj & #abc #i #d8--;,ans4 #aaaaaaaaaa & #aaaa4 ,:at is #acf #c #abi #d #adg #h #bg #cjjj8--;,ans4 #hcajj4 ,an army is made up ( #cc regim5ts ( infantry1 ea* conta9+ #hjj m52 #ad ( cavalry1 ea* conta9+ #fjj m52 & #b ( artill]y1 ea* conta9+ #cjj m54 ,! enemy has #f m regim5ts ( infantry1 ea* conta9+ #ajj m m52 #c m regim5ts ( cavalry1 ea* conta9+ #ajj m5 less2 & #d corps ( artill]y ( ! same magnitude z ^? ( ! fir/3 two regim5ts ( cavalry & "o ( infantry des]t f ! =m] to ! latt]4 ,h[ _m m5 has ! second army m ?an ! fir/8--,answ]1 #acdjj4 #fb4 ,suppose x is requir$ to multiply #bcgjg by #defg4 ,s9ce #defg is made up ( #djjj1 #ejj1 #fj1 & #g1 by 7#ec7 we mu/ multiply #bcgjg by ea* ( ^!1 & add ! products4 ,n[ 7#eh7 #bcgjg #g is #afeidi 7#fj7 #bcgjg #fj is #adbbdbj #bcgjg #ejj is #aahecejj #bcgjg #djjj is #idhbhjjj --------- ,! sum ( ^! is #ajhbfihfi :i* is ! product requir$4 ,x w d z well if1 in/ead ( writ+ ! ciph]s at ! end ( ea* l9e1 we keep ! o!r figures 9 _! places )\t !m4 ,if we take away ! ciph]s1 ! second l9e is "o place to ! left ( ! fir/1 ! ?ird "o place to ! left ( ! second1 & s on4 ,write ! multipli] & ! multiplic& ov] ^! l9es1 & ! process w /& ?us3 #bcgjg #defg ------ #afeidi #adbbdb #aahece #idhbh --------- #ajhbfihfi #fc4 ,!re is "o m case to be notic$2 t is1 :]e !re is a ciph] 9 ! middle ( ! multipli]4 ,! foll[+ example w %ew t 9 ?is case no?+ m is necess>y ?an to keep ! fir/ figure ( ea* l9e 9 ! column "u ! figure ( ! multipli] f :i* t l9e arises4 ,suppose x requir$ to multiply #cfe by #ajajja4 ,! multipli] is made up ( #ajjjjj1 #ajjj & #a4 ,proce$ z be=e1 & #cfe #a is #cfe 7#eg7 #cfe #ajjj is #cfejjj #cfe #ajjjjj is #cfejjjjj -------- ,! sum ( :i* is #cfhfecfe & ! :ole process ) ! ciph]s /ruck (f is3 #cfe #ajajja ------ #cfe #cfe #cfe -------- #cfhfecfe #fd4 ,! foll[+ is ! rule 9 all cases3 ,i4 ,place ! multipli] "u ! multiplic&1 s t ! units ( "o may be "u ^? ( ! o!r4 ,,ii4 ,multiply ! :ole multiplic& by ea* figure ( ! multipli] 7#ei71 & place ! unit ( ea* l9e 9 ! column "u ! figure ( ! multipli] f :i* x came4 ,,iii4 ,add toge!r ! l9es obta9$ by ,,ii4 column by column4 #fe4 ,:5 ! multipli] or multiplic&1 or bo?1 h ciph]s on ! "r h&1 multiply ! two toge!r )\t ! ciph]s1 & !n place on ! "r ( ! product all ! ciph]s t are on ! "r bo? ( ! multipli] & multiplic&4 ,= example1 :at is #cbjj #cjjj8 ,fir/1 #cbjj is #cb #ajj1 or "o hundr$ times z great z #cb4 ,aga91 #cb #acjjj is #cb #ac1 ) ?ree ciph]s affix$1 t is #daf1 ) ?ree ciph]s affix$1 or #dafjjj4 ,b ! product requir$ mu/ be #ajj times z great z ?is1 or mu/ h two ciph]s affix$4 ,x is !re=e #dafjjjjj1 hav+ z _m ciph]s z are 9 bo? multipli] & multiplic&4 #ff4 ,:5 any numb] is multipli$ by itself any numb] ( times1 ! result is call$ a p[] ( t numb]4 ,?us3 #f is call$ ! fir/ p[] ( #f #f #f second p[] ( #f #f #f #f ?ird p[] ( #f #f #f #f #f f\r? p[] ( #f c4 c4 ,! second & ?ird p[]s are usually call$ ! squ>e & cube1 :i* are incorrect names1 d]iv$ f c]ta9 connexions ( ! second & ?ird p[] ) ! squ>e & cube 9 geometry4 ,z ex]cises 9 multiplication1 ! foll[+ p[]s are to be f\nd4 ,numb] propos$4 ,squ>e4 ,cube4 #igb #iddghd #iahccjjdh #ajjh #ajafjfd #ajbdaibeab #cadb #ihgbafd #cajahccibhh #cafc #ajjjdefi #cafdddeagdg #eeee #cjhehjbe #agadafcbhhge #fghi #dfjijeba #cabijhedgjfi ,! fif? p[] ( #cf is #fjdffagf f\r? #ej #fbejjjj f\r? #ajh #acfjdhhif f\r? #bgg #ehhgccidda #fg4 ,x is requir$ to multiply a b by c d1 t is1 to take a b z _m times z !re are units 9 c d4 ,by 7#ec7 a b mu/ be tak5 c times1 & d times1 or ! product requir$ is 7a b7c 7a b7d4 ,b 7#eb7 7a b7c is ac bc1 & 7a b7d is ad bd2 :5ce ! product requir$ is ac bc ad bd2 or1 7a b77c d7 ac bc ad bd4 ,by simil> reason+ 7a - b77c d7 is 7a - b7c 7a - b7d2 or1 7a - b77c d7 ac - bc ad - bd4 ,to multiply a-b by c-d1 fir/ take a-b c times1 :i* gives ac-bc4 ,?is is n correct2 = 9 tak+ x c times in/ead ( c-d times1 we h tak5 x d times too _m2 or h made a result :i* is 7a-b7d too great4 ,! real result is !re=e ac-bc-7a -b7d4 ,b 7a-b7d is ad- bd1 & !re=e 7a - b77c - d7 ac - bc - ad - bd ac - bc - ad bd 7#da7 ,f ^! ?ree examples may be collect$ ! foll[+ rule = ! multiplication ( algebraic quantities3 ,multiply ea* t]m ( ! multiplic& by ea* t]m ( ! multipli]2 :5 ! two t]ms h bo? or bo?-be=e !m1 put be=e _! product2 :5 "o has & ! o!r-1 put-be=e _! product4 ,9 us+ ! fir/ t]ms1 :i* h no sign1 apply ! rule z if !y _h ! sign 4 #fh4 ,= example1 7a b77a b7 gives aa ab ab bb4 ,b ab ab is #b;ab2 h5ce ! squ>e ( a b is aa #b;ab bb4 ,aga9 7a- b77a-b7 gives aa-ab-ab bb4 ,b two subtractions ( ab are equival5t to subtract+ #b;ab2 h5ce ! squ>e ( a- b is aa-#b;ab bb4 ,aga91 7a b77a-b7 gives aa ab-ab -bb4 ,b ! addition & subtraction ( ab makes no *ange2 h5ce ! product ( a b & a- b is aa-bb4 ,aga91 ! squ>e ( a b c d or 7a b c d77a b c d7 w be f\nd to be aa #b;ab #b;ac #b;ad bb #b;bc #b;bd cc #b;cd dd2 or ! rule = squ>+ su* a quantity is3 ,squ>e ! fir/ t]m1 & multiply all t come aft] by twice t t]m2 d ! same ) ! second1 & s on to ! end4 ,,section ,,iv4 ,,division4 #fi4 ,suppose ,i ask :e!r #aef c be divid$ into a numb] ( p>ts ea* ( :i* is #ac1 or h[ _m ?irte5s #aef conta9s2 ,i propose a "q1 ! solution ( :i* is call$ ,,division4 ,9 ?is case1 #aef is call$ ! divid5d1 #ac ! divisor1 & ! numb] ( p>ts requir$ is ! quoti5t2 & :5 ,i f9d ! quoti5t1 ,i am said to divide #aef by #ac4 #gj4 ,! simple/ me?od ( do+ ?is is to subtract #ac f #aef1 & !n to subtract #ac f ! rema9d]1 & s on2 or1 9 common language1 to tell (f #aef by ?irte5s4 ,a simil> process has already occurr$ 9 ! ex]cises on subtraction1 ,art4 7#df74 ,d ?is1 & m>k "o = e subtraction t is made1 to rem9d y t ea* subtraction takes #ac once f #aef1 :i* op]ations w /& z foll[s3 #aef #ac #a ------ #adc #ac #a ------ #acj #ac #a ------ #aag #ac #a ------ #ajd #ac #a ------ #ia #ac #a ------ #gh #ac #a ------ #fe #ac #a ------ #eb #ac #a ------ #ci #ac #a ------ #bf #ac #a ------ #ac #ac #a ------ #j ,beg9 by subtract+ #ac f #aef1 :i* leaves #adc4 ,subtract #ac f #adc1 :i* leaves #acj2 & s on4 ,at la/ #ac only rema9s1 f :i* :5 #ac is subtract$1 !re rema9s no?+4 ,^u c\nt+ ! numb] ( times :i* y h subtract$ #ac1 y f9d t ?is numb] is #ab2 or #aef conta9s twelve ?irte5s1 or conta9s #ac twelve times4 ,?is me?od is ! mo/ simple possible1 & might be done ) pebbles4 ,( ^! y w\ld fir/ c\nt #aef4 ,y w\ld !n take #ac f ! heap1 & put !m into "o heap by !mselves4 ,y w\ld !n take ano!r #ac f ! heap1 & place !m 9 ano!r heap by !mselves2 & s on until !re 7 none left4 ,y w\ld !n c\nt ! numb] ( heaps1 :i* y w\ld f9d to be #ab4 #ga4 ,division is ! opposite ( multiplication4 ,9 multiplication y h a numb] ( heaps1 ) ! same numb] ( pebbles 9 ea*1 & y want to "k h[ _m pebbles !re are 9 all4 ,9 division y "k h[ _m !re are 9 all1 & h[ _m !re are to be 9 ea* heap1 & y want to "k h[ _m heaps !re are4 #gb4 ,9 ! la/ example a numb] 0 tak5 :i* conta9s an exact numb] ( ?irte5s4 ,b ?is does n happ5 ) e numb]4 ,take1 = example1 #aei4 ,foll[ ! process ( 7#gj71 & x w appe> t aft] hav+ subtract$ #ac twelve times1 !re rema9s #c1 f :i* #ac _c be subtract$4 ,we may say !n t #aei conta9s twelve ?irte5s & #c ov]2 or t #aei1 :5 divid$ by #ac1 gives a quoti5t #ab1 & a rema9d] #c4 ,if we use signs1 #aei #ac #ab #c4 ,,ex]cises4 #adf #bd #f #b1 or #adf conta9s six tw5ty-f\rs & #b ov]4 #adf #f #bd #b1 or #adf conta9s tw5ty-f\r sixes & #b ov]4 #cjj #db #g #f1 or #cjj conta9s sev5 =ty-twos & #f ov]4 #cifbd #gbgg #e #cbci4 #gc4 ,if a conta9 b q times ) a rema9d] r1 a mu/ be great] ?an bq by r2 t is1 a bq r4 ,if !re be no rema9d]1 a bq4 ,"h a is ! divid5d1 b ! divisor1 q ! quoti5t1 & r ! rema9d]4 ,9 ord] to say t a conta9s b q times1 we write1 a_/b q1 or a 3 b q1 :i* 9 old books is (t5 f\nd writt5 ?us3 a b q4 #gd4 ,if ,i divide #aef into sev]al p>ts1 & f9d h[ (t5 #ac is conta9$ 9 ea* ( !m1 x is pla9 t #aef conta9s #ac z (t5 z all its p>ts toge!r4 ,= example1 #aef is made up ( #ia1 #ci1 & #bf4 ,( ^! #ia conta9s #ac #g times1 #ci conta9s #ac #c times1 #bf conta9s #ac #b times2 !re=e #ia #ci #bf conta9s #ac #g #c #b times1 or #ab times4 ,aga91 #aef is made up ( #ajj1 #ej1 & #f4 ,n[ #ajj conta9s #ac #g times & #i ov]1 #ej conta9s #ac #c times & #aa ov]1 #f conta9s #ac #j times#i & #f ov]4 #i ,to speak always 9 ! same way1 in/ead ( say+ t #f does n conta9 #ac1 ,i say t x conta9s x #j times & #f ov]1 :i* is m]ely say+ t #f is #f m ?an no?+4 ,!re=e #ajj #ej #f conta9s #ac #g #c #j times & #i #aa #f ov]2 or #aef conta9s #ac #aj times & #bf ov]4 ,b #bf is itself #b ?irte5s2 !re=e #aef conta9s #aj ?irte5s & #b ?irte5s1 or #ab ?irte5s4 #ge4 ,! result ( ! la/ article is express$ by say+1 t if a b c d1 !n a_/m b_/m c_/m d_/m #gf4 ,9 ! fir/ example ,i did n take away #ac m ?an once at a "t1 9 ord] t ! me?od might be z simple z possible4 ,b if ,i "k :at is twice #ac1 #c times #ac1 c41 ,i c take away z _m ?irte5s at a "t z ,i please1 if ,i take c>e to m>k at ea* /ep h[ _m ,i take away4 ,= example1 take away #ac t5 times at once f #aef1 t is1 take away #acj1 & aft]w>ds take away #ac twice1 or take away #bf1 & ! process is z foll[s3 #aef #acj #aj times #ac4 --- #bf #bf #b times #ac4 --- #j ,!re=e #aef conta9s #ac #aj #b1 or #ab times4 ,aga91 to divide #cjif by #ah4 #cjif #ahjj #ajj times #ah4 ---- #abif #ijj #ej times #ah4 ---- #cif #cfj #bj times #ah4 ---- #cf #cf #b times #ah4 ---- #j ,!re=e #cjif conta9s #ah #ajj #ej #bj #b1 or #agb times4 #gg4 ,y w n[ und]/& ! foll[+ s5t5ces1 & be able to make simil> ass]tions ( o!r numb]s4 #dej is #ge #f2 x !re=e conta9s any numb]1 z #e1 #f times z (t5 z #ge conta9s x4 #ace conta9s #c m ?an #bf times2 !re=e1 ,twice #ace 0 #c 0 #eb or twice #bf times4 #aj times #ace 0 #c 0 #bfj or #aj times #bf #ej times #ace 0 #c 0 #acjj or #ej times #bf #dgb conta9s #ah m ?an #ba times2 !re=e1 #dgbj conta9s #ah m ?an #baj times1 #dgbjj conta9s #ah m ?an #bajj times1 #dgbjjj conta9s #ah m ?an #bajjj times1 #cb conta9s #ab m ?an #b times1 & less ?an #c times4 #cbj 0 #ab 0 #bj #cj #cbjj 0 #ab 0 #bjj #cjj #cbjjj 0 #ab 0 #bjjj #cjjj c4 c4 c4 #gh4 ,! =ego+ articles conta9 ! pr9ciples ( division4 ,! "q n[ is1 to apply !m 9 ! %orte/ & mo/ conv5i5t way4 ,suppose x requir$ to divide #djfh by #ah1 or to f9d #djfh_/ah 7#bc74 ,if we divide #djfh into any numb] ( p>ts1 we may1 by ! process foll[$ 9 7#gd71 f9d h[ _m times #ah is conta9$ 9 ea* ( ^! p>ts1 & f !nce h[ _m times x is conta9$ 9 ! :ole4 ,n[1 :at sep>ation ( #djfh into p>ts w be mo/ conv5i5t8 ,obs]ve t #d1 ! fir/ figure ( #djfh1 does n conta9 #ah2 b t #dj1 ! fir/ & second figures toge!r1 does conta9 #ah m ?an twice1 b less ?an ?ree times4#aj ,b #djfh 7#bj7 is made up ( #dj hundr$s1 & #fh2 ( :i*1 #dj hundr$s 7#gg7 conta9s #ah m ?an #bjj times1 & less ?an #cjj times4 ,!re=e1 #djfh also conta9s m ?an #bjj times #ah1 s9ce x mu/ conta9 #ah m times ?an #djjj does4 ,x also conta9s #ah less ?an #cjj times1 because #cjj times #ah is #edjj1 a great] numb] ?an #djfh4 ,subtract #ah #bjj times f #djfh2 t is1 subtract #cfjj1 & !re rema9s #dfh4 ,!re=e1 #djfh conta9s #ah #bjj times1 & z _m m times z #dfh conta9s #ah4 #aj ,if y h any d\bt z to ?is expression1 recollect t x means 8conta9s m ?an two eighte5s1 b n s mu* z ?ree40 ,x rema9s1 !n1 to f9d h[ _m times #dfh conta9s #ah4 ,proce$ exactly z be=e4 ,obs]ve t #df conta9s #ah m ?an twice1 & less ?an #c times2 !re=e1 #dfj conta9s x m ?an #bj1 & less ?an #cj times 7#gg72 z does also #dfh4 ,subtract #ah #bj times f #dfh1 t is1 subtract #cfj2 ! rema9d] is #ajh4 ,!re=e1 #dfh conta9s #ah #bj times1 & z _m m z #ajh conta9s x4 ,n[1 #ajh is f\nd to conta9 #ah #f times exactly2 !re=e1 #dfh conta9s x #bj #f times1 & #djfh conta9s x #bjj #bj #f times1 or #bbf times4 ,if we write d[n ! process t has be5 foll[$1 )\t any explanation1 putt+ ! divisor1 divid5d1 & quoti5t1 9 a l9e sep>at$ by p>5!ses x w /&1 z 9 example7,a74 ,let x be requir$ to divide #cfcbfeii by #acdb 7,b74 ,a4 ,b4 #ah7djfh7bjj #bj #f #acdb7cfcbfeii7bjjjj #gjjj #fj #i #cfjj #bfhdjjjj ---- -------- #dfh #idhfeii #cfj #icidjjj --- ------- #ajh #ibeii #ajh #hjebj --- ----- #j #abjgi #abjgh ----- #a ,z 9 ! previ\s example1 #cfcbfeii is sep>at$ into #cfcbjjjj & #feii2 ! fir/ f\r figures #cfcb be+ sep>at$ f ! re/1 because x takes f\r figures f ! left ( ! divid5d to make a numb] :i* is great] ?an ! divisor4 ,aga91 #cfcbjjjj is f\nd to conta9 #acdb m ?an #bjjjj1 & less ?an #cjjjj times2 & #acdb #bjjjj is subtract$ f ! divid5d1 aft] :i* ! rema9d] is #idhfeii4 ,! same op]ation is repeat$ aga9 & aga91 & ! result is f\nd to be1 t !re is a quoti5t #bjjjj #gjjj #fj #i1 or #bgjfi1 & a rema9d] #a4 ,be=e y proce$1 y %\ld n[ repeat ! =ego+ article at l5g? 9 ! solution ( ! foll[+ que/ions4 ,:at are #ajjichgd #ffggiibb #bgahbah -------- 1 -------- 1 ------- 8 #cbjg #aaddcc #acceb ! quoti5ts ( :i* are #cadg1 #ehc1 #bjc2 & ! rema9d]s #adde1 #fedhc1 #ggfb4 #gi4 ,9 ! examples ( ! la/ article1 obs]ve1 #a/1 t x is useless to write d[n ! ciph]s :i* are on ! "r ( ea* subtrah5d1 provid$ t )\t !m y keep ea* ( ! o!r figures 9 its prop] place3 #b;d1 t x is useless to put d[n ! "r h& figures ( ! divid5d s long z !y fall ov] ciph]s1 because !y d n beg9 to h any %>e 9 ! mak+ ( ! quoti5t until1 by cont9u+ ! process1 !y cease to h ciph]s "u !m3 #c;d1 t ! quoti5t is only a numb] writt5 at l5g?1 in/ead ( ! usual way4 ,= example1 ! fir/ quoti5t is #bjj #bj #f1 or #bbf2 ! second is #bjjjj #gjjj #fj #i1 or #bgjfi4 ,/rike out1 !re=e1 all ! ciph]s & ! numb]s :i* come above !m1 except ^? 9 ! fir/ l9e1 & put ! quoti5t 9 "o l9e2 & ! two examples ( ! la/ article w /& ?us3 #ah7djfh7bbf #acdb7cfcbfeii7bgjfi #cf #bfhd --- ----- #df #idhf #cf #icid --- ----- #ajh #ibei #ajh #hjeb --- ----- #j #abjgi #abjgh ----- #a #hj4 ,h5ce ! foll[+ rule is d$uc$3 ,i4 ,write ! divisor & divid5d 9 "o l9e1 & place p>5!ses on ea* side ( ! divid5d4 ,,ii4 ,take (f f ! left-h& ( ! divid5d ! lea/ numb] ( figures :i* make a numb] great] ?an ! divisor2 f9d :at numb] ( times ! divisor is conta9$ 9 ^!1 & write ?is numb] z ! fir/ figure ( ! quoti5t4 ,,iii4 ,multiply ! divisor by ! la/-m5tion$ figure1 & subtract ! product f ! numb] :i* 0 tak5 (f at ! left ( ! divid5d4 ,,iv4 ,on ! "r ( ! rema9d] place ! figure ( ! divid5d :i* comes next aft] ^? already sep>at$ 9 ,,ii43 if ! rema9d] ?us increas$ be great] ?an ! divisor1 f9d h[ _m times ! divisor is conta9$ 9 x2 put ?is numb] at ! "r ( ! fir/ figure ( ! quoti5t1 & repeat ! process3 if n1 on ! "r place ! next figure ( ! divid5d1 & ! next1 & s on until x is great]2 b rememb] to place a ciph] 9 ! quoti5t = e figure ( ! divid5d :i* y are oblig$ to take1 except ! fir/4 ,v4 ,proce$ 9 ?is way until all ! figures ( ! divid5d are exhau/$4 ,9 judg+ h[ (t5 "o l>ge numb] is conta9$ 9 ano!r1 a fir/ & r\gh guess may be made by /rik+ (f ! same numb] ( figures f bo?1 & us+ ! results in/ead ( ! numb]s !mselves4 ,?us1 #d1gcb is conta9$ 9 #ad1cgi ab\t ! same numb] ( times t #d is conta9$ 9 #ad1 or ab\t #c times4 ,! reason is1 t #d be+ conta9$ 9 #ad z (t5 z #djjj is 9 #adjjj1 & ^! la/ only diff]+ f ! propos$ numb]s by l[] d5om9ations1 viz4 hundr$s1 c4 we may expect t !re w n be mu* diff]5ce betwe5 ! numb] ( times :i* #adjjj conta9s #djjj1 & t :i* #adcgi conta9s #dgcb3 & x g5]ally happ5s s4 ,b if ! second figure ( ! divisor be #e1 or great] ?an #e1 x w be m accurate to increase ! fir/ figure ( ! divisor by #a1 be=e try+ ! me?od j expla9$4 ,no?+ b practice c give facility 9 ?is sort ( guess-"w4 #ha4 ,?is process may be made m simple :5 ! divisor is n great] ?an #ab1 if y h suffici5t k ( ! multiplication table 7#ej74 ,= example1 ,i want to divide #acbigf by #d4 ,at full l5g? ! process /&s ?us3 #d7acbigf7ccbdd #ab --- #ab #ab --- #i #h -- #ag #af --- #af #af -- #j ,b y w recollect1 )\t ! necessity ( writ+ x d[n1 t #ac conta9s #d ?ree times ) a rema9d] #a2 ?is #a y w place be=e #b1 ! next figure ( ! divid5d1 & y "k t #ab conta9s #d #c times exactly1 & s on4 ,x w be m conv5i5t to write d[n ! quoti5t ?us3 #d7acbigf ------- #ccbdd ,:ile on ?is "p ( ! subject1 we may m5tion1 t ! %orte/ way to multiply by #e is to annex a ciph] & divide by #b1 :i* is equival5t to tak+ ! half ( #aj times1 or #e times4 ,to divide by #e1 multiply by #b & /rike (f ! la/ figure1 :i* leaves ! quoti5t2 half ! la/ figure is ! rema9d]4 ,to multiply by #be1 annex two ciph]s & divide by #d4 ,to divide by #be1 multiply by #d & /rike (f ! la/ two figures1 :i* leaves ! quoti5t2 "o f\r? ( ! la/ two figures1 tak5 z "o numb]1 is ! rema9d]4 ,to multiply a numb] by #i1 annex a ciph]1 & subtract ! numb]1 :i* is equival5t to tak+ ! numb] t5 times1 & !n subtract+ x once4 ,to multiply by #ii1 annex two ciph]s & subtract ! numb]1 c4 ,9 ord] t a numb] may be divisible by #b )\t rema9d]1 its units' figure mu/ be an ev5 numb]4#aa ,t x may be divisible by #d1 its la/ two figures mu/ be divisible by #d4 ,take ! example #abcf3 ?is is compos$ ( #ab hundr$s & #cf1 ! fir/ "p ( :i*1 be+ hundr$s1 is divisible by #d1 & gives #ab tw5ty-fives2 x dep5ds !n ^u #cf1 ! la/ two figures1 :e!r #abcf is divisible by #d or n4 ,a numb] is divisible by #h if ! la/ ?ree figures are divisible by #h2 = e digit1 except ! la/ ?ree1 is a numb] ( ?\s&s1 & #ajjj is divisible by #h2 :e!r !re=e ! :ole %all be divisible by #h or n dep5ds on ! la/ ?ree figures3 ?us1 #abgidf is n divisible by #h1 s9ce #idf is n s4 ,a numb] is divisible by #c or #i only :5 ! sum ( its digits is divisible by #c or #i4 ,take = example #abcd2 ?is is #aa ,among ! ev5 figures we include #j4 #a ?\s&1 or #iii & #a #b hundr$1 or twice #ii & #b #c t5s1 or ?ree times #i & #c & #d or #d ,n[ #i1 #ii1 #iii1 c4 are all obvi\sly divisible by #i & by #c1 & s w be any numb] made by ! repetition ( all or any ( !m any numb] ( times4 ,x !re=e dep5ds on #a #b #c #d1 or ! sum ( ! digits1 :e!r #abcd %all be divisible by #i or #c1 or n4 ,f ! above we ga!r1 t a numb] is divisible by #f :5 x is ev51 & :5 ! sum ( its digits is divisible by #c4 ,la/ly1 a numb] is divisible by #e only :5 ! la/ figure is #j or #e4 #hb4 ,:]e ! divisor is unity foll[$ by ciph]s1 ! rule becomes extremely simple1 z y w see by ! foll[+ examples3 #ajj7ccdbi7ccd #cjj ---- #cdb #cjj ---- #dbi #djj --- #bi ,?is is1 !n1 ! rule3 ,cut (f z _m figures f ! "r h& ( ! divid5d z !re are ciph]s4 ,^! figures w be ! rema9d]1 & ! re/ ( ! divid5d w be ! quoti5t4 #aj7bgagcaf -------- #bgagca & rem4 #f4 ,or we may prove ^! results ?us3 f 7#bj71 #bgagcaf is #bgagca t5s & #f2 ( :i* ! fir/ conta9s #aj #bgagca times1 & ! second n at all2 ! quoti5t is !re=e #bgagca1 & ! rema9d] #f 7#gb74 ,aga9 7#bj71 #ccdbi is #ccd hundr$s & #bi2 ( :i* ! fir/ conta9s #ajj #ccd times1 & ! second n at all2 ! quoti5t is !re=e #ccd1 & ! rema9d] #bi4 #hc4 ,! foll[+ examples w %ew h[ ! rule may be %ort5$ :5 !re are ciph]s 9 ! divisor4 ,) ea* example is plac$ ano!r conta9+ ! same process1 all unnecess>y figures be+ remov$2 & f ! comp>ison ( ! two1 ! rule at ! end ( ?is article is d]iv$4 ,i4 #aghbjjj7fdbdgjjjjj7cfje #aghb7fdbdgjj7cfje #ecdfjjj #ecdf -------- ---- #ajghgjjj #ajghg #ajfibjjj #ajfib ---------- ------- #iejjjjj #iejj #hiajjjj #hiaj ------- ------- #eijjjj #eijjjj ,,ii4 #abcjjjjj7dbagfahicjj7cdbh #abc7dbagfa7cdbh #cfijjjjj #cfi --------- ---- #ebgfahic #ebg #dibjjjjj #dib --------- ---- #cefahicj #cef #bdfjjjjj #bdf --------- ---- #aajahicjj #aaja #ihdjjjjj #ihd -------- ---------- #aaghicjj #aaghicjj ,! rule1 !n1 is3 ,/rike out z _m figures#ab f ! "r ( ! divid5d z !re are ciph]s at ! "r ( ! divisor4 ,/rike out all ! ciph]s f ! divisor1 & divide 9 ! usual way2 b at ! end ( ! process place on ! "r ( ! rema9d] all ^? figures :i* 7 /ruck out ( ! divid5d4 #ab ,includ+ bo? ciph]s & o!rs4 #hd4 ,,ex]cises4 ,divid5d4 ,divisor4 ,quoti5t4,rema9d]4 #ifid #dg #bjf #ab #agefah #cacf #ef #b #bcgifdhd #acjjjj #ahc #fdhd #adjjbefd #ahga #gdhd #j #cajcaddbj #ghgh #cicij #j #cicijdjfdg #fhhi #egaghg #d #bbhgfgibdedifa #dcjdfgba #ecadda #j ,%ew t #ajj #ajj #ajj - #dc #dc #dc ,i4 ------------------------------ #ajj #ajj #ajj #dc #dc #dc4 #ajj - #dc #ajj #ajj #ajj #dc #dc #dc ,,ii4 ------------------------------ #ajj #ajj - #ajj #dc #dc #dc4 #ajj #dc #gf #gf #b #gf #eb #eb #eb ,,iii4 -------------------------------- #gf #eb4 #gf #eb #ab #ab #ab #ab - #a ,,iv4 #a #ab #ab #ab #ab #ab #ab ----------------------4 #ab - #a ,:at is ! ne>e/ numb] to #acgfdbi :i* c be divid$ by #cfcjj )\t rema9d]8--,answ]1 #acgidjj4 ,if #cf ox5 c eat #baf acres ( grass 9 "o ye>1 & if a %eep eat half z mu* z an ox1 h[ long w x take #di ox5 & #acf %eep toge!r to eat #ageej acres8--,answ]1 #be ye>s4 #he4 ,take any two numb]s1 "o ( :i* divides ! o!r )\t rema9d]2 = example1 #cb & #d4 ,multiply bo? ^! numb]s by any o!r numb]2 = example1 #f4 ,! products w be #aib & #bd4 ,n[1 #aib conta9s #bd j z (t5 z #cb conta9s #d4 ,suppose #f baskets1 ea* conta9+ #cb pebbles1 ! :ole numb] ( :i* w be #aib4 ,take #d f "o basket1 "t aft] "t1 until t basket is empty4 ,x is pla9 t if1 in/ead ( tak+ #d f t basket1 ,i take #d f ea*1 ! :ole #f w be empti$ toge!r3 t is1 #f times #cb conta9s #f times #d j z (t5 z #cb conta9s #d4 ,! same reason+ applies to o!r numb]s1 & !re=e we d n alt] ! quoti5t if we multiply ! divid5d & divisor by ! same numb]4 #hf4 ,aga91 suppose t #bjj is to be divid$ by #ej4 ,divide bo? ! divid5d & divisor by ! same numb]2 = example1 #e4 ,!n1 #bjj is #e times #dj1 & #ej is #e times #aj4 ,b by 7#he71 #dj divid$ by #aj gives ! same quoti5t z #e times #dj divid$ by #e times #aj1 & !re=e ! quoti5t ( two numb]s is n alt]$ by divid+ bo? ! divid5d & divisor by ! same numb]4 #hg4 ,f 7#ee71 if a numb] be multipli$ successively by two o!rs1 x is multipli$ by _! product4 ,?us1 #bg1 fir/ multipli$ by #e1 & ! product multipli$ by #c1 is ! same z #bg multipli$ by #e times #c1 or #ae4 ,also1 if a numb] be divid$ by any numb]1 & ! quoti5t be divid$ by ano!r1 x is ! same z if ! fir/ numb] _h be5 divid$ by ! product ( ! o!r two4 ,= example1 divide #fj by #d1 :i* gives #ae1 & ! quoti5t by #c1 :i* gives #e4 ,x is pla91 t if ea* ( ! f\r fifte5s ( :i* #fj is compos$ be divid$ into ?ree equal p>ts1 !re are twelve equal p>ts 9 all2 or1 a division by #d1 & !n by #c1 is equival5t to a division by #d #c1 or #ab4 #hh4 ,! foll[+ rules w be bett] und]/ood by /at+ !m 9 an example4 ,if #cb be multipli$ by #bd & divid$ by #f1 ! result is ! same z if #cb _h be5 multipli$ by ! quoti5t ( #bd divid$ by #f1 t is1 by #d2 = ! six? "p ( #bd be+ #d1 ! six? "p ( any numb] repeat$ #bd times is t numb] repeat$ #d times2 or1 multiply+ by #bd & divid+ by #f is equival5t to multiply+ by #d4 #hi4 ,aga91 if #dh be multipli$ by #d1 & t product be divid$ by #bd1 x is ! same ?+ z if #dh 7 divid$ at once by ! quoti5t ( #bd divid$ by #d1 t is1 by #f4 ,=1 e unit :i* is repeat$ #f times 9 #dh is repeat$ #d times z (t51 or #bd times1 9 #d times #dh1 or ! quoti5t ( #dh & #f is ! same z ! quoti5t ( #dh #d & #f #d4 #ij4 ,! results ( ! la/ five articles may be algebraically express$ ?us3 ma a ---- ---- 7#he7 mb b ,if n divide a & b )\t rema9d]1 a ---- n a ------ ---- 7#hf7 b b ---- n a ---- b a ------ ---- 7#hg7 c bc ab b ------ a ---- 7#hh7 c c ac a ----- ------ 7#hi7 b b ---- c ,x mu/ be recollect$1 h[ev]1 t ^! h only be5 prov$ 9 ! case :]e all ! divisions are )\t rema9d]4 #ia4 ,:5 "o numb] divides ano!r )\t leav+ any rema9d]1 or is conta9$ an exact numb] ( times 9 x1 x is said to be a measure ( t numb]1 or to measure x4 ,?us1 #d is a measure ( #acf1 or measures #acf2 b x does n measure #acg4 ,! reason = us+ ! ^w measure is ?is3 ,suppose y h a rod #d feet long1 ) no?+ m>k$ ^u x1 ) :i* y want to measure "s l5g?2 = example1 ! l5g? ( a /reet4 ,if t /reet %\ld happ5 to be #acf feet 9 l5g?1 y w be able to measure x ) ! rod1 because1 s9ce #acf conta9s #d #cd times1 y w f9d t ! /reet is exactly #cd times ! l5g? ( ! rod4 ,b if ! /reet %\ld happ5 to be #acg feet long1 y _c measure x ) ! rod2 = :5 y h measur$ #cd ( ! rods1 y w f9d a rema9d]1 ^: l5g? y _c tell )\t "s %ort] measure4 ,h5ce #d is said to measure #acf1 b n to measure #acg4 ,a measure1 !n1 is a divisor :i* leaves no rema9d]4 #ib4 ,:5 "o numb] is a measure ( two o!rs1 x is call$ a common measure ( ! two4 ,?us1 #ae is a common measure ( #ahj & #ge4 ,two numb]s may h sev]al common measures4 ,= example1 #cfj & #afh h ! common measures #b1 #c1 #d1 #f1 #bd1 & sev]al o!rs4 ,n[1 ?is "q maybe ask$3 ,( all ! common measures ( #cfj & #afh1 :i* is ! greate/8 ,! answ] to ?is "q is d]iv$ f a rule ( ari?metic1 call$ ! rule = f9d+ ! ,,greate/ ,,common ,,measure1 :i* we proce$ to consid]4 #ic4 ,if "o quantity measure two o!rs1 x measures _! sum & diff]5ce4 ,?us1 #g measures #ba & #ef4 ,x !re=e measures #ef #ba & #ef-ba1 or #gg & #ce4 ,?is is only ano!r way ( say+ :at 0 said 9 7#gd74 #id4 ,if "o numb] measure a second1 x measures e numb] :i* ! second measures4 ,?us1 #e measures #ae1 & #ae measures #cj1 #de1 #fj1 #ge1 c42 all :i* numb]s are measur$ by #e4 ,x is pla9 t if #ae conta9s #e #c times1 #cj1 or #ae #ae conta9s #e #c #c times1 or #f times1 #de1 or #ae #ae #ae conta9s #e #c #c #c or #i times2 & s on4 #ie4 ,e numb] :i* measures bo? ! divid5d & divisor measures ! rema9d] also4 ,to %ew ?is1 divide #cfj by #aab4 ,! quoti5t is #c1 & ! rema9d] #bd1 t is 7#gb7 #cfj is ?ree times #aab & #bd1 or #cfj #aab #c #bd4 ,f ?is x foll[s1 t #bd is ! diff]5ce betwe5 #cfj & #c times #aab1 or #bd #cfj-aab #c4 ,take any numb] :i* measures bo? #cfj & #aab2 = example1 #d4 ,!n #d measures #cfj1 #d measures #aab1 & !re=e 7#id7 measures #aab #c1 or #aab #aab #aab4 ,!re=e 7#ic7 x measures #cfj-aab #c1 :i* is ! rema9d] #bd4 ,! same reason+ may be appli$ to all o!r measures ( #cfj & #aab2 & ! result is1 t e quantity :i* measures bo? ! divid5d & divisor also measures ! rema9d]4 ,h5ce1 e common measure ( a divid5d & divisor is also a common measure ( ! divisor & rema9d]4 #if4 ,e common measure ( ! divisor & rema9d] is also a common measure ( ! divid5d & divisor4 ,take ! same example1 & recollect t #cfj #aab #c #bd4 ,take any common measure ( ! rema9d] #bd & ! divisor #aab2 = example1 #h4 ,!n #h measures #bd2 & #h measures #aab1 & !re=e 7#id7 measures #aab #c4 ,!re=e 7#ic7 #h measures #aab #c #bd1 or measures ! divid5d #cfj4 ,!n e common measure ( ! rema9d] & divisor is also a common measure ( ! divisor & divid5d1 or !re is no common measure ( ! rema9d] & divisor :i* is n also a common measure ( ! divisor & divid5d4 #ig4 ,i4 ,x is prov$ 9 7#ie7 t ! rema9d] & divisor h all ! common measures :i* are 9 ! divid5d & divisor4 ,,ii4 ,x is prov$ 9 7#if7 t !y h no o!rs4 ,x !re=e foll[s1 t ! greate/ ( ! common measures ( ! fir/ two is ! greate/ ( ^? ( ! second two1 :i* %ews h[ to f9d ! greate/ common measure ( any two numb]s1#ac z foll[s3 #ih4 ,take ! prec$+ example1 & let x be requir$ to f9d ! g4 c4 m4 ( #cfj & #aab1 & obs]ve t #cfj divid$ by #aab gives ! rema9d] #bd1 #aab divid$ by #bd gives ! rema9d] #af1 #bd divid$ by #af gives ! rema9d] #h1 #af divid$ by #h gives no rema9d]4 #ac ,= %ortness1 ,i abbreviate ! words greate/ common measure into _! initial lett]s1 g4 c4 m4 ,n[1 s9ce #h divides #af )\t rema9d]1 & s9ce x also divides itself )\t rema9d]1 #h is ! g4 c4 m4 ( #h & #af1 because x is impossible to divide #h by any numb] great] ?an #h2 s t1 ev5 if #af _h a great] measure ?an #h1 x c\ld n be common to #af & #h4 ,!re=e #h is g4 c4 m4 ( #af & #h1 7#ig7 g4 c4 m4 ( #af & #h is g4 c4 m4 ( #bd & #af1 g4 c4 m4 ( #bd & #af is g4 c4 m4 ( #aab & #bd1 g4 c4 m4 ( #aab & #bd is g4 c4 m4 ( #cfj & #aab1 ,!re=e #h is g4 c4 m4 ( #cfj & #aab4 ,! process c>ri$ on may be writt5 d[n 9 ei!r ( ! foll[+ ways3 #aab7cfj7c #ccf --- #bd7aab7d #aab #cfj #c #if #if #ccf #d --- ----------- #af7bd7a #af #bd #a #af #af #af #b -- ----------- #h7af7b #j #h #af -- #j ,! rule = f9d+ ! greate/ common measure ( two numb]s is1 ,i4 ,divide ! great] ( ! two by ! less4 ,,ii4 ,make ! rema9d] a divisor1 & ! divisor a divid5d1 & f9d ano!r rema9d]4 ,,iii4 ,proce$ 9 ?is way until !re is no rema9d]1 & ! la/ divisor is ! greate/ common measure requir$4 #ii4 ,y may p]haps ask h[ ! rule is to %ew :5 ! two numb]s h no common measure4 ,! fact is1 t !re are1 /rictly speak+1 no su* numb]s1 because all numb]s are measur$ by #a2 t is1 conta9 an exact numb] ( units1 & !re=e #a is a common measure ( e two numb]s4 ,if !y h no o!r common measure1 ! la/ divisor w be #a1 z 9 ! foll[+ example1 :]e ! greate/ common measure ( #hg & #be is f\nd4 #be7hg7c #ge -- #ab7be7b #bd -- #a7ab7ab #ab -- #j ,,ex]cises4 ,numb]s4 g4 c4 m4 #faig #ieba #a #ehcfc #bfjb #a #eedg #adgjjhddc #ahdi #fbha #cbfjda #ega #bhiae #cadie #e #aeji #cjjcji #c ,:at are #cf #cf #b #cf #gb #gb #gb & #cf #cf #cf #gb #gb #gb2 & :at is _! greate/ common measure8--,answ]1 #aaffd4 #ajj4 ,if two numb]s be divisible by a ?ird1 & if ! quoti5ts be aga9 divisible by a f\r?1 t ?ird is n ! greate/ common measure4 ,= example1 #cfj & #ejd are bo? divisible by #d4 ,! quoti5ts are #ij & #abf4 ,n[ #ij & #abf are bo? divisible by #i1 ! quoti5ts ( :i* division are #aj & #ad4 ,by 7#hg71 divid+ a numb] by #d1 & !n divid+ ! quoti5t by #i1 is ! same ?+ z divid+ ! numb] itself by #d #i1 or by #cf4 ,!n1 s9ce #cf is a common measure ( #cfj & #ejd1 & is great] ?an #d1 #d is n ! greate/ common measure4 ,aga91 s9ce #aj & #ad are bo? divisible by #b1 #cf is n ! greate/ common measure4 ,x !re=e foll[s1 t :5 two numb]s are divid$ by _! greate/ common measure1 ! quoti5ts h no common measure except #a 7#ii74 ,o!rwise1 ! numb] :i* 0 call$ ! greate/ common measure 9 ! la/ s5t5ce is n s 9 reality4 #aja4 ,to f9d ! greate/ common measure ( ?ree numb]s1 f9d ! g4 c4 m4 ( ! fir/ & second1 & ( ?is & ! ?ird4 ,= s9ce all common divisors ( ! fir/ & second are conta9$ 9 _! g4 c4 m41 & no o!rs1 :atev] is common to ! fir/1 second1 & ?ird1 is common also to ! ?ird & ! g4 c4 m4 ( ! fir/ & second1 & no o!rs4 ,simil>ly1 to f9d ! g4 c4 m4 ( f\r numb]s1 f9d ! g4 c4 m4 ( ! fir/1 second1 & ?ird1 & ( t & ! f\r?4 #ajb4 ,:5 a fir/ numb] conta9s a second1 or is divisible by x )\t rema9d]1 ! fir/ is call$ a multiple ( ! second4 ,! words multiple & measure are ?us connect$3 ,s9ce #d is a measure ( #bd1 #bd is a multiple ( #d4 ,! numb] #if is a multiple ( #h1 #ab1 #bd1 #dh1 & sev]al o!rs4 ,x is !re=e call$ a common multiple ( #h1 #ab1 #bd4 #dh1 c4 ,! product ( any two numb]s is evid5tly a common multiple ( bo?4 ,?us1 #cf #h1 or #bhh1 is a common multiple ( #cf & #h4 ,b !re are common multiples ( #cf & #h less ?an #bhh2 & because x is conv5i5t1 :5 a common multiple ( two quantities is want$1 to use ! lea/ ( !m1 ,i n[ %ew h[ to f9d ! lea/ common multiple ( two numb]s4 #ajc4 ,take1 = example1 #cf & #h4 ,f9d _! greate/ common measure1 :i* is #d1 & obs]ve t #cf is #i #d1 & #h is #b #d4 ,! quoti5ts ( #cf & #h1 :5 divid$ by _! greate/ common measure1 are !re=e #i & #b4 ,multiply ^! quoti5ts toge!r1 & multiply ! product by ! greate/ common measure1 #d1 :i* gives #i #b #d1 or #gb4 ,?is is a multiple ( #h1 or ( #d #b by 7#ee72 & also ( #cf or ( #d #i4 ,x is also ! lea/ common multiple2 b ?is _c be prov$ to y1 because ! demon/ration _c be ?or\ghly und]/ood )\t m practice 9 ! use ( lett]s to /& = numb]s4 ,b y may satisfy y\rself t x is ! lea/ 9 ?is case1 & t ! same process w give ! lea/ common multiple 9 any o!r case :i* y may take4 ,x is n ev5 necess>y t y %\ld "k x is ! lea/4 ,:5ev] a common multiple is to be us$1 any "o w d z well z ! lea/4 ,x is only to avoid l>ge numb]s t ! lea/ is us$ 9 pref]5ce to any o!r4 ,:5 ! greate/ common measure is #a1 ! lea/ common multiple ( ! two numb]s is _! product4 ,! rule !n is3 ,to f9d ! lea/ common multiple ( two numb]s1 f9d _! greate/ common measure1 & multiply "o ( ! numb]s by ! quoti5t :i* ! o!r gives :5 divid$ by ! greate/ common measure4 ,to f9d ! lea/ common multiple ( ?ree numb]s1 f9d ! lea/ common multiple ( ! fir/ two1 & f9d ! lea/ common multiple ( t multiple & ! ?ird1 & s on4 ,,ex]cises4 ,numb]s propos$4 ,lea/ common multiple4 #ad1 #ba #db #af1 #e1 #bd #bdj #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #aj #bebj #f1 #h1 #aa1 #af1 #bj #bfdj #hgf1 #hfd #fcjgb #hfh1 #hed #ebidh ,a conv5i5t mode ( f9d+ ! lea/ common multiple ( sev]al numb]s is z foll[s1 :5 ! common measures are easily visible3 ,pick out a numb] ( common measures ( two or m1 :i* h !mselves no divisors great] ?an unity4 ,write !m z divisors1 & divide e numb] :i* w divide by "o or m ( !m4 ,br+ d[n ! quoti5ts1 & also ! numb]s :i* w n divide by any ( !m4 ,repeat ! process ) ! results1 & s on until ! numb]s br\ght d[n h no two ( !m any common measure except unity4 ,!n1 = ! lea/ common multiple1 multiply all ! divisors by all ! numb]s la/ br\ght d[n4 ,= in/ance1 let x be requir$ to f9d ! lea/ common multiple ( all ! numb]s f #aa to #ba4 #b1 #b1 #c1 #e1 #g7aa #ab #ac #ad #ae #af #ag #ah #ai #bj #ba --------------------------------- #aa #a #ac #a #a #d #ag #c #ai #a #a ,!re are n[ no common measures left 9 ! r[1 & ! lea/ common multiple requir$ is ! product ( #b1 #b1 #c1 #e1 #g1 #aa1 #ac1 #d1 #ag1 #c1 & #ai2 or #bcbgibefj4 ,,section ,v4 ,,fractions4 #ajd4 ,suppose x requir$ to divide #di y>ds into five equal p>ts1 or1 z x is call$1 to f9d ! fif? "p ( #di y>ds4 ,if we divide #de by #e1 ! quoti5t is #i1 & ! rema9d] is #d2 t is 7#gb71 #di is made up ( #e times #i & #d4 ,let ! l9e ,a ,b repres5t #di y>ds3 ,a-------------------------------------- --,b ,c-------------- ,i -- ,d-------------- ,k -- ,e-------------- ,l -- ,f-------------- ,m -- ,g-------------- ,n -- ,i ,k ,l ,m ,n ,h ----- ,take #e l9es1 ,c1 ,d1 ,e1 ,f1 & ,g1 ea* #i y>ds 9 l5g?1 & ! l9e ,h1 #d y>ds 9 l5g?4 ,!n1 s9ce #di is #e n9es & #d1 ,c1 ,d1 ,e1 ,f1 ,g1 & ,h1 are toge!r equal to ,a ,b4 ,divide ,h1 :i* is #d y>ds1 into five equal p>ts1 ,i1 ,k1 ,l1 ,m1 & ,n1 & place "o ( ^! p>ts opposite to ea* ( ! l9es1 ,c1 ,d1 ,e1 ,f1 & ,g4 ,x foll[s t ! t5 l9es1 ,c1 ,d1 ,e1 ,f1 ,g1 ,i1 ,k1 ,l1 ,m1 ,n1 are toge!r equal to ,a ,b1 or #di y>ds4 ,n[ ,d & ,k toge!r are ( ! same l5g? z ,c & ,i toge!r1 & s are ,e & ,l1 ,f & ,m1 & ,g & ,n4 ,!re=e1 ,c & ,i toge!r1 repeat$ #e times1 w be #di y>ds2 t is1 ,c & ,i toge!r make up ! fif? "p ( #di y>ds4 #aje4 ,c is a c]ta9 numb] ( y>ds1 viz4 #i2 b ,i is a new sort ( quantity1 to :i* hi!rto we h nev] come4 ,x is n an exact numb] ( y>ds1 = x arises f divid+ #d y>ds into #e p>ts1 & tak+ "o ( ^? p>ts4 ,x is ! fif? "p ( #d y>ds1 & is call$ a ,,fraction ( a y>d4 ,x is writt5 ?us1 _/7#bc71 & is :at we mu/ add to #i y>ds 9 ord] to make up ! fif? "p ( #di y>ds4 ,! same reason+ w\ld apply to divid+ #di bu%els ( corn1 or #di acres ( l&1 into #e equal p>ts4 ,we %\ld f9d = ! fif? "p ( ! fir/1 #i bu%els & ! fif? "p ( #d bu%els2 & = ! second1 #i acres & ! fif? "p ( #d acres4 ,we say1 !n1 once = all1 t ! fif? "p ( #di is #i & _/1 or #i _/2 :i* is usually writt5 7#i_/71 or if we use signs1 #di_/e 7#i_/74 ,,ex]cises4 ,:at is ! sev5te5? "p ( #abcg8--;,answ]1 7#gb-_/74 #ajjcb #ffchai #bbggccii ,:at are -----1 ------1 & -------- 8 #aigd #bcgaj #bdbd #afb #bcfdi #bcdc ,answ]1 7#e ----71 7#bg -----71 7#icid ----74 #aigd #bcgaj #bdbd #ajf4 ,by ! t]m fraction is und]/ood a "p ( any numb]1 or ! sum ( any ( ! equal p>ts into :i* a numb] is divid$4 ,?us1 _/1 _/1 _/1 are fractions4 ,! t]m fraction ev5 includes :ole numb]s3#ad = example1 #ag is _/1 _/1 _/1 c4 #ad ,numb]s :i* conta9 an exact numb] ( units1 su* z #e1 #g1 #ajj1 c41 are call$ :ole numb]s or integ]s1 :5 we wi% to di/+ui% !m f fractions4 ,! upp] numb] is call$ ! num]ator1 ! l[] numb] is call$ ! d5om9ator1 & bo? ( ^! are call$ t]ms ( ! fraction4 ,z long z ! num]ator is less ?an ! d5om9ator1 ! fraction is less ?an a unit3 ?us1 _/ is less ?an a unit2 = #f divid$ into #f p>ts gives #a = ea* "p1 & mu/ give less :5 divid$ into #ag p>ts4 ,simil>ly1 ! fraction is equal to a unit :5 ! num]ator & d5om9ator are equal1 & great] ?an a unit :5 ! num]ator is great] ?an ! d5om9ator4 #ajg4 ,by is meant ! ?ird "p ( #b4 ,?is is ! same z twice ! ?ird "p ( #a4 ,to prove ?is1 let ,a ,b be two y>ds1 & divide ea* ( ! y>ds ,a ,c & ,c ,b into ?ree equal p>ts4 ------------ ,a ,d ,e ,c ,f ,g ,b ,!n1 because ,a ,e1 ,e ,f1 & ,f ,b1 are all equal to "o ano!r1 ,a ,e is ! ?ird "p ( #b4 ,x is !re=e 4 ,b ,a ,e is twice ,a ,d1 & ,a ,d is ! ?ird "p ( "o y>d1 or 2 !re=e is twice 2 t is1 9 ord] to get ! l5g? 1 x makes no diff]5ce :e!r we divide two y>ds at once into ?ree p>ts1 & take "o ( !m1 or :e!r we divide "o y>d into ?ree p>ts1 & take two ( !m4 ,by ! same reason+1 may be f\nd ei!r by divid+ #e into #h p>ts1 & tak+ "o ( !m1 or by divid+ #a into #h p>ts1 & tak+ five ( !m4 ,9 future1 ( ^! two mean+s ,i %all use t :i* is mo/ conv5i5t at ! "t1 z x is prov$ t !y are ! same ?+4 ,?is pr9ciple is ! same z ! foll[+3 ,! ?ird "p ( any numb] may be obta9$ by add+ toge!r ! ?irds ( all ! units ( :i* x consi/s4 ,?us1 ! ?ird "p ( #b1 or ( two units1 is made by tak+ "o-?ird out ( ea* ( ! units1 t is1 #b4 ,?is mean+ appe>s ambigu\s :5 ! num]ator is great] ?an ! d5om9ator3 ?us1 _/ w\ld mean t #a is to be divid$ into #g p>ts1 & #ae ( !m are to be tak54 ,we %\ld "h let z _m units be ea* divid$ into #g p>ts z w give m ?an #ae ( ^? p>ts1 & take #ae ( !m4 #ajh4 ,! value ( a fraction is n alt]$ by multiply+ ! num]ator & d5om9ator by ! same quantity4 ,take ! fraction 1 multiply its num]ator & d5om9ator by #e1 & x becomes _/1 :i* is ! same ?+ z 2 t is1 "o-tw5tie? "p ( #ae y>ds is ! same ?+ z "o-f\r? ( #c y>ds3 or1 if our second mean+ ( ! ^w fraction be us$1 y get ! same l5g? by divid+ a y>d into #bj p>ts & tak+ #ae ( !m1 z y get by divid+ x into #d p>ts & tak+ #c ( !m4 ,to prove ?is1 ,illu/ration let ,a ,b repres5t a y>d2 divide x into #d equal p>ts1 ,a ,c1 ,c ,d1 ,d ,e1 & ,e ,b1 & divide ea* ( ^! p>ts into #e equal p>ts4 ,!n ,a ,e is 4 ,b ! second division cuts ! l9e into #bj equal p>ts1 ( :i* ,a ,e conta9s #ae4 ,x is !re=e _/4 ,!re=e1 _/ & are ! same ?+4 ,aga91 s9ce is made f _/ by divid+ bo? ! num]ator & d5om9ator by #e1 ! value ( a fraction is n alt]$ by divid+ bo? its num]ator & d5om9ator by ! same quantity4 ,?is pr9ciple1 :i* is ( s mu* importance 9 e "p ( ari?metic1 is (t5 us$ 9 common language1 z :5 we say t #ad out ( #ba is #b out ( #c1 c4 #aji4 ,?\gh ! two fractions & _/ are ! same 9 value1 & ei!r ( !m may be us$ = ! o!r )\t error1 yet ! fir/ is m conv5i5t ?an ! second1 n only because y h a cle>] idea ( ! f\r? ( ?ree y>ds ?an ( ! tw5tie? "p ( fifte5 y>ds1 b because ! numb]s 9 ! fir/ be+ small]1 are m conv5i5t = multiplication & division4 ,x is !re=e useful1 :5 a fraction is giv51 to f9d out :e!r its num]ator & d5om9ator h any common divisors or common measures4 ,9 7#ih7 0 giv5 a rule = f9d+ ! greate/ common measure ( any two numb]s2 & x 0 %ewn t :5 ! two numb]s are divid$ by _! greate/ common measure1 ! quoti5ts h no common measure except #a4 ,f9d ! greate/ common measure ( ! t]ms ( ! fraction1 & divide !m by t numb]4 ,! fraction is !n said to be r$uc$ to its l[e/ t]ms1 & is 9 ! /ate 9 :i* ! be/ notion c be =m$ ( its magnitude4 ,,ex]cises4 ,) ea* fraction is writt5 ! same r$uc$ to its l[e/ t]ms4 #bgid #bb #abg #bb ---- ---------- ---- #biba #bc #abg #bc #bghh #ag #afd #ag ---- ---------- ---- #dibj #cj #afd #cj #icbjh #gfd #abb #gfd ----- ---------- --- #acghf #aac #abb #aac #hhhhjj #bb #djdjj #bb -------- ------------ ----- #djceifjj #iii #djdjj #iii #iedfi #aba #ghi #aba ------ ----------- --- #ceighd #def #ghi #def #aaj4 ,:5 ! t]ms ( ! fraction giv5 are already 9 factors1#ae any "o factor 9 ! num]ator may be divid$ by a numb]1 provid$ "s "o factor 9 ! d5om9ator is divid$ by ! same4 ,?is foll[s f 7#hh7 & 7#ajh74 ,9 ! foll[+ examples ! figures alt]$ by division are acc5t$4 #ae ,a factor ( a numb] is a numb] :i* divides x )\t rema9d]3 ?us1 #d1 #f1 #h1 are factors ( #bd1 & #f #d1 #h #c1 #b #b #b #c1 are sev]al ways ( decompos+ #bd into factors4 #ab #aa #aj #c #aa #aj #a #aa #e ------------ ------------- ------------- #ee4 #b #c #d #b #c #a #a #a #a #ah #ae #ac #b #c #a #a #a #a ------------ ------------- ------------- _/4 #bj #ed #eb #d #f #d #b #b #d #bg #bh #c #d #c #b ------- --------- -------- _/4 #i #gj #a #aj #a #e #aaa4 ,z we c1 by 7#ajh71 multiply ! num]ator & d5om9ator ( a fraction by any numb]1 )\t alt]+ its value1 we c n[ readily r$uce two fractions to two o!rs1 :i* %all h ! same value z ! fir/ two1 & :i* %all h ! same d5om9ator4 ,take1 = example1 & _/2 multiply bo? t]ms ( by #g1 & bo? t]ms ( _/ by #c4 ,x !n appe>s t #b #g is ------- or _/ #c #g #d #c _/ is ------- or _/4 #g #c ,"h are !n two fractions _/ & _/1 equal to & _/1 & hav+ ! same d5om9ator1 #ba2 9 ?is case1 & _/ are said to be r$uc$ to a common d5om9ator4 ,x is requir$ to r$uce 1 1 & _/ to a common d5om9ator4 ,multiply bo? t]ms ( ! fir/ by ! product ( #f & #i2 ( ! second by ! product ( #aj & #i2 & ( ! ?ird by ! product ( #aj & #f4 ,!n x appe>s 7#ajh7 t #a #f #i is ----------- or _/ #aj #f #i #e #aj #i is ------------ or _/ #f #aj #i #g #aj #f _/ is ------------ or _/4 #i #aj #f ,on look+ at ^! la/ fractions1 we see t all ! num]ators & ! common d5om9ator are divisible by #f1 & 7#ajh7 ?is division w n alt] _! values4 ,on divid+ ! num]ators & d5om9ators ( _/1 _/1 & _/ by #f1 ! result+ fractions are1 _/1 _/1 & _/4 ,^! are fractions ) a common d5om9ator1 & :i* are ! same z 1 1 & _/2 & !re=e ^! are a m simple answ] to ! "q ?an ! fir/ fractions4 ,obs]ve also t #edj is "o common multiple ( #aj1 #f1 & #i1 namely1 #aj #f #i1 b t #ij is ! lea/ common multiple ( #aj1 #f1 & #i 7#ajc74 ,! foll[+ process1 !re=e1 is bett]4 ,to r$uce ! fractions 1 1 & _/1 to o!rs hav+ ! same value & a common d5om9ator1 beg9 by f9d+ ! lea/ common multiple ( #aj1 #f1 & #i1 by ! rule 9 7#ajc71 :i* is #ij4 ,obs]ve t #aj1 #f1 & #i are conta9$ 9 #ij #i1 #ae1 & #aj times4 ,multiply bo? t]ms ( ! fir/ by #i1 ( ! second by #ae1 & ( ! ?ird by #aj1 & ! fractions ?us produc$ are _/1 _/1 & _/1 ! same z be=e4 ,if "o ( ! numb]s be a :ole numb]1 x may be r$uc$ to a fraction hav+ ! common d5om9ator ( ! re/1 by 7#ajf74 ,,ex]cises4 ,fractions propos$ r$uc$ to a common d5om9ator4 #b #a #a #bj #f #e --- --- --- ---- --- --- #c #e #f #cj #cj #cj #a #b #c #ab #c #bh #bd #ah #dh #fc --- --- --- ---- --- --- --- --- --- --- #c #g #ad #ba #d #hd #hd #hd #hd #hd #d #e #f #cjjj #djj #ej #f #c --- ---- ---- ---- ----- ----- ---- #aj #ajj #ajjj #ajjj #ajjj #ajjj #ajjj #cc #bha #bbcda #ajfdii ---- ---- -------- ------ #cgi #fgg #befehc #befehc #aab4 ,by r$uc+ two fractions to a common d5om9ator1 we are able to comp>e !m2 t is1 to tell :i* is ! great] & :i* ! less ( ! two4 ,= example1 take & _/4 ,^! fractions r$uc$1 )\t alt]ation ( _! value1 to a common d5om9ator1 are _/ & _/4 ,( ^! ! fir/ mu/ be ! great]1 because 7#ajg7 x may be obta9$ by divid+ #a into #cj equal p>ts & tak+ #ae ( !m1 b ! second is made by tak+ #ad ( ^? p>ts4 ,x is evid5t t ( two fractions :i* h ! same d5om9ator1 ! great] has ! great] num]ator2 & also t ( two fractions :i* h ! same num]ator1 ! great] has ! less d5om9ator4 ,?us1 _/ is great] ?an _/1 s9ce ! fir/ is a #g?1 & ! la/ only a #i? "p ( #h4 ,also1 any num]ator may be made to belong to z small a fraction z we please1 by suffici5tly increas+ ! d5om9ator4 ,?us1 _/ is _/1 _/ is _/1 & _/ is _/ 7#ajh74 ,we c n[ also increase & dim9i% ! fir/ fraction by ! second4 ,= ! fir/ fraction is made up ( #ae ( ! #cj equal p>ts into :i* #a is divid$4 ,! second fraction is #ad ( ^? p>ts4 ,! sum ( ! two1 !re=e1 mu/ be #ae #ad1 or #bi ( ^? p>ts2 t is1 _/ is _/4 ,! diff]5ce ( ! two mu/ be #ae-ad1 or #a ( ^? p>ts2 t is1 -_/ _/4 #aac4 ,f ! la/ two articles ! foll[+ rules are obta9$3 ,i4 ,to comp>e1 to add1 or to subtract fractions1 fir/ r$uce !m to a common d5om9ator4 ,:5 ?is has be5 done1 t is ! greate/ ( ! fractions :i* has ! greate/ num]ator4 ,_! sum has ! sum ( ! num]ators = its num]ator1 & ! common d5om9ator = its d5om9ator4 ,_! diff]5ce has ! diff]5ce ( ! num]ators = its num]ator1 & ! common d5om9ator = its d5om9ator4 ,,ex]cises4 #a #a #a #a #ec #dd #aec #ahcbi --- --- --- - --- ---- ---- - ----- ------- #b #c #d #e #fj #c #dbg #abhb #h #c #d #ahcd #a #ab #bec #a ---- ---- ---- ---- #b - --- ---- ---- #aj #ajj #ajjj #ajjj #g #ac #ia #a #h #id #c #afc #ig #icjff --- --- ---- --- --- - ---- ------- #b #af #ahh #b #eba #hha #deijja #aad4 ,suppose x requir$ to add a :ole numb] to a fraction1 = example1 #f to _/4 ,by 7#ajf7 #f is _/1 & _/ _/ is _/2 t is1 #f _/1 or z x is usually writt51 7#f_/71 is _/4 ,! rule 9 ?is case is3 ,multiply ! :ole numb] by ! d5om9ator ( ! fraction1 & to ! product add ! num]ator ( ! fraction2 ! sum w be ! num]ator ( ! result1 & ! d5om9ator ( ! fraction w be its d5om9ator4 ,?us1 7#c7 _/1 7#bb_/7 _/1 7#gd_/7 _/4 ,?is rule is ! opposite ( t 9 7#aje74 #aae4 ,f ! la/ rule x appe>s t #ijg #agbcjijg #bbe #ffgbbe #agbc ------ is --------1 #ffg ----- is ------1 #ajjjj #ajjjj #ajjj #ajjj #ii #bcjjjii & #bc ------ is -------4 #ajjjj #ajjjj ,h5ce1 :5 a :ole numb] is to be add$ to a fraction ^: d5om9ator is #a foll[$ by ciph]s1 ! numb] ( :i* is n less ?an ! numb] ( figures 9 ! num]ator1 ! rule is3 ,write ! :ole numb] fir/1 & !n ! num]ator ( ! fraction1 ) z _m ciph]s betwe5 !m z ! numb] ( ciph]s 9 ! d5om9ator exce$s ! numb] ( figures 9 ! num]ator4 ,?is is ! num]ator ( ! result1 & ! d5om9ator ( ! fraction is its d5om9ator4 ,if ! numb] ( ciph]s 9 ! d5om9ator be equal to ! numb] ( figures 9 ! num]ator1 write no ciph]s betwe5 ! :ole numb] & ! num]ator4 ,,ex]cises4 ,r$uce ! foll[+ mix$ quantities to fractions3 #bcgjg #f #bii #bbaj #a ------1 #bdeg ---1 #abjg --------1 & #bcc -----4 #ajjjjj #aj #ajjjjjjj #ajjjj #aaf4 ,suppose x requir$ to multiply by #d4 ,?is by 7#dh7 is tak+ f\r times2 t is1 f9d+ 4 ,?is by 7#aab7 is _/2 s t to multiply a fraction by a :ole numb] ! rule is3 ,multiply ! num]ator by ! :ole numb]1 & let ! d5om9ator rema94 #aag4 ,if ! d5om9ator ( ! fraction be divisible by ! :ole numb]1 ! rule may be /at$ ?us3 ,divide ! d5om9ator ( ! fraction by ! :ole numb]1 & let ! num]ator rema94 ,= example1 multiply _/ by #f4 ,?is 7#aaf7 is _/1 :i*1 s9ce ! num]ator & d5om9ator are n[ divisible by #f1 is 7#ajh7 ! same z _/4 ,x is pla9 t _/ is made f _/ 9 ! mann] /at$ 9 ! rule4 #aah4 ,multiplication has be5 def9$ to be ! tak+ z _m ( "o numb] z !re are units 9 ano!r4 ,?us1 to multiply #ab by #g is to take z _m twelves z !re are units 9 #g1 or to take #ab z _m times z y mu/ take #a 9 ord] to make #g4 ,?us1 :at is done ) #a 9 ord] to make #g1 is done ) #ab to make #g times #ab4 ,= example1 #g is #a #a #a #a #a #a #a #g times #ab is #ab #ab #ab #ab #ab #ab #ab4 ,:5 ! same ?+ is done ) two fractions1 ! result is /ill call$ _! product1 & ! process is /ill call$ multiplication4 ,!re is ?is diff]5ce1 t :]eas a :ole numb] is made by add+ #a to itself a numb] ( times1 a fraction is made by divid+ #a into a numb] ( equal p>ts1 & add+ "o ( ^! p>ts to itself a numb] ( times4 ,?is be+ ! mean+ ( ! ^w multiplication1 z appli$ to fractions1 :at is multipli$ by 8 ,:atev] is done ) #a 9 ord] to make mu/ n[ be done ) 2 b to make 1 #a is divid$ into #h p>ts1 & #g ( !m are tak54 ,!re=e1 to make 1 mu/ be divid$ into #h p>ts1 & #g ( !m mu/ be tak54 ,n[ is1 by 7#ajh71 ! same ?+ z _/4 ,s9ce _/ is made by divid+ #a into #cb p>ts1 & tak+ #bd ( !m1 or1 :i* is ! same ?+1 tak+ #c ( !m #h times1 if _/ be divid$ into #h equal p>ts1 ea* ( !m is _/2 & if #g ( ^! p>ts be tak51 ! result is _/ 7#aaf73 !re=e multipli$ by is _/2 & ! same reason+ may be appli$ to any o!r fractions4 ,b _/ is made f & by multiply+ ! two num]ators toge!r = ! num]ator1 & ! two d5om9ators = ! d5om9ator2 :i* furni%es a rule = ! multiplication ( fractions4 #aai4 ,if ?is product _/ is to be multipli$ by a ?ird fraction1 = example1 by _/1 ! result is1 by ! same rule1 _/2 & s on4 ,! g5]al rule = multiply+ any numb] ( fractions toge!r is !re=e3 ,multiply all ! num]ators toge!r = ! num]ator ( ! product1 & all ! d5om9ators toge!r = its d5om9ator4 #abj4 ,suppose x requir$ to multiply toge!r _/ & _/4 ,! product may be writt5 ?us3 #ae #h #abj -------1 & is ----1 #af #aj #afj :i* r$uc$ to its l[e/ t]ms 7#aji7 is 4 ,?is result might h be5 obta9$ directly1 by obs]v+ t #ae & #aj are bo? measur$ by #e1 & #h & #af are bo? measur$ by #h1 & t ! fraction may be writt5 ?us3 #c #e #h -------------4 #b #h #b #e ,divide bo? its num]ator & d5om9ator by #e #h 7#ajh7 & 7#hg71 & ! result is at once 2 !re=e1 be=e proce$+ to multiply any numb] ( fractions toge!r1 if !re be any num]ator & any d5om9ator1 :e!r belong+ to ! same fraction or n1 :i* h a common measure1 divide !m bo? by t common measure1 & use ! quoti5ts in/ead ( ! divid5ds4 ,a :ole numb] may be consid]$ z a fraction ^: d5om9ator is #a2 ?us1 #af is _/ 7#ajf72 & ! same rule w apply :5 "o or m ( ! quantities are :ole numb]s4 ,,ex]cises #acf #bfh #cfddh #ahbbd ---- --- ------- ------- #gdgj #iai #fhfdicj #cdcbdfe #a #b #c #d #a #b #ag #b --- --- --- --- --- ---- ---- ---- #b #c #d #e #e #ag #de #de #b #ac #bda #fbff #ac #fja #ghac --- ---- ---- ------ ---- ---- ----- #ei #g #ai #ghdg #dfa #aa #ejga ,fraction propos$4 ,squ>e4 ,cube4 #gja #diadja #cdddgbaja ----- ------- --------- #aeh #bdifd #ciddcab #adj #aifjj #bgddjjj ----- ------ -------- #ada #aihha #bhjcbba #cee #abfjbe #ddgchhge ----- ------- --------- #aac #abgfi #addbhig ,f #ajj acres ( gr\nd1 two-?irds ( !m are tak5 away2 #ej acres are !n add$ to ! result1 & _/ ( ! :ole is tak52 :at numb] ( acres does ?is produce8--,answ]1 7#ei_/74 #aba4 ,9 divid+ "o :ole numb] by ano!r1 = example1 #ajh by #i1 ?is "q is ask$1--,c we1 by ! addition ( any numb] ( n9es1 produce #ajh8 & if s1 h[ _m n9es w be suffici5t = t purpose8 ,suppose we take two fractions1 = example1 & 1 & ask1 ,c we1 by divid+ into "s numb] ( equal p>ts1 & add+ a numb] ( ^! p>ts toge!r1 produce 8 if s1 into h[ _m p>ts mu/ we divide 1 & h[ _m ( !m mu/ we add toge!r8 ,! solution ( ?is "q is /ill call$ ! division ( by 2 & ! fraction ^: d5om9ator is ! numb] ( p>ts into :i* is divid$1 & ^: num]ator is ! numb] ( !m :i* is tak51 is call$ ! quoti5t4 ,! solution ( ?is "q is z foll[s3 ,r$uce bo? ^! fractions to a common d5om9ator 7#aaa71 :i* does n alt] _! value 7#ajh72 !y !n become _/ & _/4 ,! "q n[ is1 to divide _/ into a numb] ( p>ts1 & to produce _/ by tak+ a numb] ( ^! p>ts4 ,s9ce _/ is made by divid+ #a into #ae p>ts & tak+ #ab ( !m1 if we divide _/ into #ab equal p>ts1 ea* ( ^! p>ts is _/2 if we take #aj ( ^! p>ts1 ! result is _/4 ,!re=e1 9 ord] to produce _/ or 7#ajh71 we mu/ divide _/ or into #ab p>ts1 & take #aj ( !m2 t is1 ! quoti5t is _/4 ,if we call ! divid5d1 & ! divisor1 z be=e1 ! quoti5t 9 ?is case is d]iv$ f ! foll[+ rule1 :i* ! same reason+ w %ew to apply to o!r cases3 ,! num]ator ( ! quoti5t is ! num]ator ( ! divid5d multipli$ by ! d5om9ator ( ! divisor4 ,! d5om9ator ( ! quoti5t is ! d5om9ator ( ! divid5d multipli$ by ! num]ator ( ! divisor4 ,?is rule is ! rev]se ( multiplication1 z w be se5 by comp>+ :at is requir$ 9 bo? cases4 ,9 multiply+ by _/1 ,i ask1 if out ( be tak5 #aj p>ts out ( #ab1 h[ mu* ( a unit is tak51 & ! answ] is _/1 or 4 ,aga91 9 divid+ by 1 ,i ask :at "p ( is 1 ! answ] to :i* is _/4 #abb4 ,by tak+ ! foll[+ in/ance1 we %all see t ?is rule c be sometimes simplifi$4 ,divide _/ by _/4 ,obs]ve t #af is #d #d1 & #bh is #d #g2 #cc is #c #aa1 & #ae is #c #e2 !re=e ! two fractions are #d #d #d #g ------ & -----1 #c #aa #c #e & _! quoti5t1 accord+ to ! rule1 is #d #d #c #e --------------1 #c #aa #d #g 9 :i* #d #c is f\nd bo? 9 ! num]ator & d5om9ator4 ,! fraction is !re=e 7#ajh7 ! same z #d #e #bj ------1 or ----4 #aa #g #gg ,! rule ( ! la/ article1 !re=e1 admits ( ?is modification3 ,if ! two num]ators or ! two d5om9ators h a common measure1 divide by t common measure1 & use ! quoti5ts in/ead ( ! divid5ds4 #abc4 ,9 divid+ a fraction by a :ole numb]1 = example1 by #ae1 consid] #ae z ! fraction _/4 ,! rule gives _/ z ! quoti5t4 ,!re=e1 to divide a fraction by a :ole numb]1 multiply ! d5om9ator by t :ole numb]4 ,,ex]cises4 ,divid5d4 ,divisor4 ,quoti5t4 #da #fc #da ---- ---- ----- #cc #aa #ahi #dfg #ijg #dgafg ---- ---- ------- #aea #aja #acfieg #ghac #fja #ac ----- ---- ---- #ejga #aa #dfa _/ _/ _/ - _/ _/ _/ ,:at are -----------------------------------1 _/ - _/ & _/ _/ - _/ _/ ----------------------- 8 _/ - _/ #eei ,answ]1 ----1 & #a4 #gbbe ,a c reap a field 9 #ab days1 ,b 9 #f1 & ,c 9 #d days2 9 :at "t c !y all d x toge!r8#af--;,answ]1 #b days4 #af ,! me?od ( solv+ ?is & ! foll[+ "q may be %ewn ?us3 ,if ! numb] ( days 9 :i* ea* c\ld reap ! field is giv51 ! "p :i* ea* c\ld d 9 a day by himself c be f\nd1 & !nce ! "p :i* all c\ld d toge!r2 ?is be+ kn[n1 ! numb] ( days :i* x w\ld take all to d ! :ole c be f\nd4 ,9 :at "t w\ld a ci/]n be fill$ by cocks :i* w\ld sep>ately fill x 9 #ab1 #aa1 #aj1 & #i h\rs8--,answ]1 7#b_/7 h\rs4 #abd4 ,! pr9cipal results ( ?is section may be exhibit$ algebraically z foll[s2 let a1 b1 c1 c4 /& = any :ole numb]s4 ,!n a #a a ma 7#ajg7 ---- ---- a 7#ajh7 ---- ---- b b b mb a c ad bc 7#aaa7 --- & --- are ! same z ---- & ---- b d bd bd a b a b 7#aab7 --- --- --------- c c c a b a - b --- - --- --------- c c c a c ad bc 7#aac7 --- --- ----------- b d bd a c ad - bc --- - --- ----------- b d bd a c ac 7#aah7 --- --- ---- b d bd a c a_/b ad 7#aba7 --- div4 by --- or --------- ---- b d c_/d bc #abe4 ,^! results are true ev5 :5 ! lett]s !mselves repres5t fractions4 ,= example1 take ! fraction a_/b -------1 c_/d ^: num]ator & d5om9ator are fractional1 & multiply its num]ator & d5om9ator by ! fraction e ae_/bf ---1 :i* gives ----------1 f ce_/df a$f :i* 7#aba7 is -------1 bfce :i*1 divid+ ! num]ator & d5om9ator by ef 7#ajh71 is ad ----4 bc ,b ! orig9al fraction itself is ad a_/b 7a_/b7 7e_/f7 ----2 h5ce ------- --------------------- bc c_/d 7c_/d7 7e_/f7 :i* corresponds to ! second =mula#ag 9 7#abd74 ,9 a simil> mann] x may be %ewn1 t ! o!r =mul ( ! same article are true :5 ! lett]s !re us$ ei!r repres5t fractions1 or are remov$ & fractions introduc$ 9 _! place4 ,all =mul e/abli%$ ?r\gh\t ?is "w are equally true :5 fractions are sub/itut$ = :ole numb]s4 ,= example 7#ed71 7m n7a ma na4 ,let m1 n1 & a be respectively ! fractions p r b ---1 ---1 & ---4 q s c ,!n m n is p r ps qr --- ---1 or ----------- q s qs & 7m n7a is ps qr b 7ps qr7b ----------- ---1 or ---------------- qs c qsc psb qrb or -------------4 qsc psb qrb pb rb ,b ?is 7#aab7 is ----- -----1 :i* is ---- ----1 qsc qsc qc sc psb pb qrb rb s9ce ----- ----1 & ----- ---- 7#ajc74 qsc qc qsc sc pb p b rb r b ,b ---- --- ---1 & ---- --- ---4 qc q c sc s c ,!re=e 7m n7a1 or 7p r 7b p b r b 7--- --- 7--- --- --- --- ---4 7q s 7c q c s c ,9 a simil> mann] ! same may be prov$ ( any o!r =mula4 #ag ,a =mula is a "n giv5 to any algebraical expression :i* is commonly us$4 ,! foll[+ examples may be useful3 a c e g --- --- --- --- b d f h acfh bdeg --------------------- --------------- a e c g a$h bcfg --- --- --- --- b f d h #a b ---------- --------- #a ab #a a --- b #a #a bc #a --------------- -------------- ----------------- #a c abc a c a --------- a -------- #a bc #a b --- c #a #a #eg ,?us1 ------------ --------- --- #a #h #cej #f ------- #f --- #a #eg #g --- #h ,! rules t h be5 prov$ to hold good = all numb]s may be appli$ :5 ! numb]s are repres5t$ by lett]s4 ,,section ,,vi4 ,,decimal ,,fractions4 #abf4 ,we h se5 7#aab7 7#aba7 ! necessity ( r$uc+ fractions to a common d5om9ator1 9 ord] to comp>e _! magnitudes4 ,we h se5 also h[ mu* m readily op]ations are p]=m$ ^u fractions :i* h ! same1 ?an ^u ^? :i* h diff]5t1 d5om9ators4 ,on ?is acc\nt x has long be5 cu/om>y1 9 all ^? p>ts ( ma!matics :]e fractions are (t5 requir$1 to use none b su* z ei!r h1 or c be easily r$uc$ to o!rs hav+1 ! same d5om9ators4 ,n[1 ( all numb]s1 ^? :i* c be mo/ easily manag$ are su* z #aj1 #ajj1 #ajjj1 c41 :]e #a is foll[$ by ciph]s4 ,^! are call$ ,,decimal ,,numb]s2 & a fraction ^: d5om9ator is any "o ( !m1 is call$ a ,,decimal ,,fraction1 or m commonly1 a ,,decimal4 #abg4 ,a :ole numb] may be r$uc$ to a decimal fraction1 or "o decimal fraction to ano!r1 ) ! greate/ ease4 ,= example1 #idj #idjj #idjjj #id is ---1 or ----1 or ----- 7#ajf72 #aj #ajj #ajjj #c #cj #cjj #cjjj ---- is ----1 or ----1 or ----- 7#ajh74 #cj #ajj #ajjj #ajjjj ,! plac+ ( a ciph] on ! "r h& ( any numb] is ! same ?+ z multiply+ t numb] by #aj 7#eg71 & ?is may be done z (t5 z we please 9 ! num]ator ( a fraction1 provid$ x be done z (t5 9 ! d5om9ator 7#ajh74 #abh4 ,! next "q is1 ,h[ c we r$uce a fraction :i* is n decimal to ano!r :i* is1 )\t alt]+ its value8 ,take1 = example1 ! fraction _/1 multiply bo? ! num]ator & d5om9ator successively by #aj1 #ajj1 #ajjj1 c41 :i* w give a s]ies ( fractions1 ea* ( :i* is equal to _/ 7#ajh71 viz4 _/1 _/1 _/1 _/1 c4 ,! d5om9ator ( ea* ( ^! fractions c be divid$ )\t rema9d] by #af1 ! quoti5ts ( :i* divisions =m ! s]ies ( decimal numb]s #aj1 #ajj1 #ajjj1 #ajjjj1 c4 ,if1 !re=e1 "o ( ! num]ators be divisible by #af1 ! fraction to :i* t num]ator belongs has a num]ator & d5om9ator bo? divisible by #af4 ,:5 t division has be5 made1 :i* 7#ajh7 does n alt] ! value ( ! fraction1 we %all h a fraction ^: d5om9ator is "o ( ! s]ies #aj1 #ajj1 #ajjj1 c41 & :i* is equal 9 value to _/4 ,! "q is !n r$uc$ to f9d+ ! fir/ ( ! numb]s #gj1 #gjj1 #gjjj1 #gjjjj1 c41 :i* c be divid$ by #af )\t rema9d]4 ,divide ^! numb]s1 "o aft] ! o!r1 by #af1 z foll[s3 #af7gj7d #af7gjj7dc #af7gjjj7dcg #af7gjjjj7dcge #fd #fd #fd #fd -- --- --- --- #f #fj #fj #fj #dh #dh #dh -- --- --- #ab #abj #abj #aab #aab --- --- #h #hj #hj -- #j ,x appe>s1 !n1 t #gjjjj is ! fir/ ( ! num]ators :i* is divisible by #af4 ,b x is n necess>y to write d[n ea* ( ^! divisions1 s9ce x is pla9 t ! la/ conta9s all :i* came be=e4 ,x w d1 !n1 to proce$ at once z if ! numb] ( ciph]s 7 )\t end1 to /op :5 ! rema9d] is no?+1 & !n c\nt ! numb] ( ciph]s :i* h be5 us$4 ,9 ?is case1 s9ce #gjjjj is #af #dcge1 #gjjjj #af #dcge #dcge ------1 :i* is ----------1 or -----1 #afjjjj #af #ajjjj #ajjjj gives ! fraction requir$4 ,!re=e1 to r$uce a fraction to a decimal fraction1 annex ciph]s to ! num]ator1 & divide by ! d5om9ator until !re is no rema9d]4 ,! quoti5t w be ! num]ator ( ! requir$ fraction1 & ! d5om9ator w be unity1 foll[$ by z _m ciph]s z 7 us$ 9 obta9+ ! quoti5t4 ,,ex]cises4 ,r$uce to decimal fractions 1 1 _/1 _/1 _/1 & _/4 ,answ]1 _/1 _/1 _/1 _/1 _/1 & _/4 #abi4 ,x w happ5 9 mo/ cases t ! annex+ ( ciph]s to ! num]ator w nev] make x divisible by ! d5om9ator )\t rema9d]4 ,= example1 try to r$uce _/ to a decimal fraction4 #g7ajjjjjjjjjjjjjjjjjj1 c4 ------------------- #adbhegadbhegadbheg1 c4 ,! quoti5t "h is a cont9ual repetition ( ! figures #a1 #d1 #b1 #h1 #e1 #g1 9 ! same ord]2 !re=e _/ _c be r$uc$ to a decimal fraction4 ,b1 nev]!less1 if we take z a num]ator any numb] ( figures f ! quoti5t #adbhegadbheg1 c41 & z a d5om9ator #a foll[$ by z _m ciph]s z 7 us$ 9 mak+ t "p ( ! quoti5t1 we %all get a fraction :i* diff]s v little f _/1 & :i* w diff] /ill less f x if we put m figures 9 ! num]ator & m ciph]s 9 ! d5om9ator4 ,?us1 #a is less #a #c :i* is n #a --- --- by --- --- #aj ?an #g #gj s mu* z #aj #ad #a #b #a --- --- --- --- #ajj #g #gjj #ajj #adb #a #f #a ---- --- ---- ---- #ajjj #g #gjjj #ajjj #adbh #a #d #a ----- --- ----- ----- #ajjjj #g #gjjjj #ajjjj #adbhe #a #e #a ------ --- ------ ------ #ajjjjj #g #gjjjjj #ajjjjj #adbheg #a #a #a ------- --- ------- ------- #ajjjjjj #g #gjjjjjj #ajjjjjj c4 c4 c4 c4 ,9 ! fir/ column is a s]ies ( decimal fractions1 :i* come ne>] & ne>] to _/1 z ! ?ird column %ews4 ,!re=e1 ?\gh we _c f9d a decimal fraction :i* is exactly _/1 we c f9d "o :i* diff]s f x z little z we please4 ,?is may also be illu/rat$ ?us3 ,x is requir$ to r$uce _/ to a decimal fraction )\t ! error ( say a million? ( a unit2 multiply ! num]ator & d5om9ator ( _/ by a million1 & !n divide bo? by #g2 we h !n #a #ajjjjjj #adbhega_/ --- ------- ----------- #g #gjjjjjj #ajjjjjj ,if we reject ! fraction _/ 9 ! num]ator1 :at we reject is really ! #g? "p ( ! million? "p ( a unit2 or less ?an ! million? "p ( a unit4 ,!re=e _/ is ! fraction requir$4 ,,ex]cises4 ,make simil> tables #c #ag #a ) ---1 ---1 & ---4 ^! fractions #ia #adc #bdg #c ,! recurr+ --- is #cbifgj1cbifgj1 c4 quoti5t ( #ia #ag --- #aahhha1aahhha1 c4 #adc #a --- #djdhehbiieieadagjj1djdhehb c4 #bdg #acj4 ,! reason = ! recurr5ce ( ! figures ( ! quoti5t 9 ! same ord] is z foll[s3 ,if #ajjj1 c4 be divid$ by ! numb] #bdg1 ! rema9d] at ea* /ep ( ! division is less ?an #bdg1 be+ ei!r #j1 or "o ( ! fir/ #bdf numb]s4 ,if1 !n1 ! rema9d] nev] become no?+1 by c>ry+ ! division f> en\gh1 "o rema9d] w occur a second "t4 ,if possible1 let ! fir/ #bdf rema9d]s be all diff]5t1 t is1 let !m be #a1 #b1 #c1 c41 up to #bdf1 v>i\sly di/ribut$4 ,z ! #bdg? rema9d] _c be s great z #bdg1 x mu/ be "o ( ^! :i* h prec$$4 ,f ! /ep :]e ! rema9d] becomes ! same z a =m] rema9d]1 x is evid5t t =m] figures ( ! quoti5t mu/ be repeat$ 9 ! same ord]4 #aca4 ,y w "h naturally ask1 ,:at is ! use ( decimal fractions1 if ! great] numb] ( fractions _c be r$uc$ at all to decimals8 ,! answ] is ?is3 ,! addition1 subtraction1 multiplication1 & division ( decimal fractions are mu* easi] ?an ^? ( common fractions2 & ?\gh we _c r$uce all common fractions to decimals1 yet we c f9d decimal fractions s ne> to ea* ( !m1 t ! error aris+ f us+ ! decimal in/ead ( ! common fraction w n be p]ceptible4 ,= example1 if we suppose an in* to be divid$ into t5 million ( equal p>ts1 "o ( ^? p>ts by itself w n be visible to ! eye4 ,!re=e1 9 f9d+ a l5g?1 an error ( a t5-million? "p ( an in* is ( no consequ5ce1 ev5 :]e ! f9e/ measurem5t is necess>y4 ,n[1 by c>ry+ on ! table 9 7#abi71 we %all see t #adbhega #a #a -------- does n diff] f --- by --------2 #ajjjjjjj #g #ajjjjjjj & if ^! fractions repres5t$ p>ts ( an in*1 ! fir/ might be us$ = ! second1 s9ce ! diff]5ce is n p]ceptible4 ,9 apply+ ari?metic to practice1 no?+ c be measur$ s accurately z to be repres5t$ 9 numb]s )\t any error :atev]1 :e!r x be l5g?1 weight1 or any o!r species ( magnitude4 ,x is !re=e unnecess>y to use any o!r ?an decimal fractions1 s9ce1 by means ( !m1 any quantity may be repres5t$ ) z mu* correctness z by any o!r me?od4 ,,ex]cises4 ,f9d decimal fractions :i* d n diff] f ! foll[+ fractions by _/4 ,answ]1 _/4 _/ _/4 _/ _/4 _/ _/4 #acb4 ,e decimal may be imm$iately r$uc$ to a quantity consi/+ ei!r ( a :ole numb] & m simple decimals1 or ( m simple decimals alone1 hav+ "o figure only 9 ea* ( ! num]ators4 ,take1 = example1 #adgcbf #adgcbf #cbf ------4 ,by 7#aae7 ------- is #adg----2 #ajjj #ajjj #ajjj & s9ce #cbf is made up ( #cjj1 & #bj1 & #f2 by 7#aab7 _/ _/ _/ _/4 ,b 7#ajh7 _/ is _/1 & _/ is _/4 ,!re=e1 #f_/ is made up ( #adg _/ _/ #f_/4 ,n[1 take any numb]1 = example1 #adgcbf1 & =m a numb] ( fractions hav+ = _! num]ators ?is numb]1 & = _! d5om9ators #a1 #aj1 #ajj1 #ajjj1 #ajjjj1 c41 & r$uce ^! fractions into numb]s & m simple decimals1 9 ! =ego+ mann]1 :i* w give ! table bel[4 ,,decomposition ,,( ,a ,,decimal ,,fraction4 #adgcbf ------ #adgcbf #a #adgcbf #f ------ #adgcb --- #aj #aj #adgcbf #b #f ------ #adgc --- --- #ajj #aj #ajj #adgcbf #c #b #f ------ #adg --- --- ---- #ajjj #aj #ajj #ajjj #adgcbf #g #c #b #f ------ #ad --- --- ---- ----- #ajjjj #aj #ajj #ajjj #ajjjj #adgcbf #d #g #c #b #f ------ #a --- --- ---- ----- ------ #ajjjjj #aj #ajj #ajjj #ajjjj #ajjjjj #adgcbf #a #d #g #c #b #f ------- --- --- ---- ----- ------ ------- #ajjjjjj #aj #ajj #ajjj #ajjjj #ajjjjj #ajjjjjj #adgcbf #a #d #g #c #b #f -------- --- ---- ----- ------ ------- -------- #ajjjjjjj #ajj #ajjj #ajjjj #ajjjjj #ajjjjjj #ajjjjjjj ,n4,b4 ,! /ud5t %\ld write ?is table himself1 & !n proce$ to make simil> tables f ! foll[+ ex]cises4 ,,ex]cises4 ,r$uce ! foll[+ fractions into a s]ies ( numb]s & m simple fractions3 #cadaeibf #cadaeibf --------1 --------1 c4 #aj #ajj #bgjjjca #bgjjjca -------1 --------1 c4 #aj #ajj #bjgcjjj #bjgcjjj -------1 --------1 c4 #aj #ajj #cccacjc #cccacjc -------1 -------1 c4 #ajjj #ajjjj #acc4 ,if1 9 ?is table1 & o!rs made 9 ! same mann]1 y look at ^? fractions :i* conta9 a :ole numb]1 y w see t !y may be made ?us3 ,m>k (f1 f ! "r h& ( ! num]ator1 z _m figures z !re are ciph]s 9 ! d5om9ator by a po9t1 or any o!r conv5i5t m>k4 ,?is w give #adgcbf :5 ! fraction is #adgcbf ------ #aj #adgcbf #adgcbf ------ #ajj #adgcbf #adgcbf ------ #ajjj c4 c4 ,! figures on ! left ( ! po9t by !mselves make ! :ole numb] :i* ! fraction conta9s4 ,( ^? on its "r1 ! fir/ is ! num]ator ( ! fraction ^: d5om9ator is #aj1 ! second ( t ^: d5om9ator is #ajj1 & s on4 ,we n[ come to ^? fractions :i* d n conta9 a :ole numb]4 #acd4 ,! fir/ ( ^! is _/ :i* ! numb] ( ciph]s 9 ! d5om9ator is ! same z ! numb] ( figures 9 ! num]ator4 ,if we /ill foll[ ! same rule1 & m>k (f all ! figures1 by plac+ ! po9t be=e !m all1 ?us1 #adgcbf1 ! obs]vation 9 7#acc7 /ill holds good2 =1 on look+ at _/ 9 ! table1 we f9d x is #a #d #g #c #b #f --- --- ---- ----- ------ ------- #aj #ajj #ajjj #ajjjj #ajjjjj #ajjjjjj ,! next fraction is _/1 :i* we f9d by ! table to be #a #d #g #c #b #f --- ---- ----- ------ ------- -------- #ajj #ajjj #ajjjj #ajjjjj #ajjjjjj #ajjjjjjj ,9 ?is1 #a is n divid$ by #aj1 b by #ajj2 if1 !re=e1 we put a po9t be=e ! :ole1 ! rule is n true1 = ! fir/ figure on ! left ( ! po9t has ! d5om9ator :i*1 accord+ to ! rule1 ! second ought to h1 ! second t :i* ! ?ird ought to h1 & s on4 ,9 ord] to keep ! same rule = ?is case1 we mu/ contrive to make #a ! second figure on ! "r ( ! po9t in/ead ( ! fir/4 ,?is may be done by plac+ a ciph] betwe5 x & ! po9t1 ?us1 #jadgcbf4 ,"h ! rule holds good1 = by t rule ?is fraction is #j #a #d #g #c #b #f --- --- ---- ----- ------ ------- -------- #aj #ajj #ajjj #ajjjj #ajjjjj #ajjjjjj #ajjjjjjj :i* is ! same z ! prec$+ l9e1 s9ce _/ is #j1 & ne$ n be reckon$4 ,simil>ly1 :5 !re are two ciph]s m 9 ! d5om9ator ?an !re are figures 9 ! num]ator1 ! rule w be true if we place two ciph]s betwe5 ! po9t & ! num]ator4 ,! rule1 !re=e1 /at$ fully1 is ?is3 ,to r$uce a decimal fraction to a :ole numb] & m simple decimals1 or to m simple decimals alone if x d n conta9 a :ole numb]1 m>k (f by a po9t z _m figures f ! num]ator z !re are ciph]s 9 ! d5om9ator4 ,if ! num]ator h n places en\gh = ?is1 write z _m ciph]s be=e x z x wants places1 & put ! po9t be=e ^! ciph]s4 ,!n1 if !re be any figures be=e ! po9t1 !y make ! :ole numb] :i* ! fraction conta9s4 ,! fir/ figure aft] ! po9t ) ! d5om9ator #aj1 ! second ) ! d5om9ator #ajj1 & s on1 are ! fractions ( :i* ! fir/ fraction is compos$4 #ace4 ,decimal fractions are n usually writt5 at full l5g?4 ,x is m conv5i5t to write ! num]ator only1 & to cut (f f ! num]ator z _m figures z !re are ciph]s 9 ! d5om9ator1 :5 t is possible1 by a po9t4 ,:5 !re are m ciph]s 9 ! d5om9ator ?an figures 9 ! num]ator1 z _m ciph]s are plac$ be=e ! num]ator z w supply ! defici5cy1 & ! po9t is plac$ be=e ! ciph]s4 ,?us1 #g w be us$ 9 future to d5ote _/1 #jg = _/1 & s on4 ,! foll[+ tables w give ! :ole ( ?is notation at "o view1 & w %ew its connexion ) ! decimal notation expla9$ 9 ! fir/ section4 ,y w obs]ve t ! numb]s on ! "r ( ! units' place /& = units divid$ by #aj1 #ajj1 #ajjj1 c4 :ile ^? on ! left are units multipli$ by #aj1 #ajj1 #ajjj1 c4 ,! /ud5t is recomm5d$ always to write ! decimal po9t 9 a l9e ) ! top ( ! figures or 9 ! middle1 z is done "h1 & nev] at ! bottom4 ,! reason is1 t x is usual 9 ! high] bran*es ( ma!matics to use a po9t plac$ betwe5 two numb]s or lett]s :i* are multipli$ toge!r2 ?us1 #ae4 #af1 a4 b1 7a b74 7c d7 /& = ! products ( ^? numb]s or lett]s4 #abcd #d #d ,i4 #abcd /&s = ---- or #abc--- or #abc --- #aj #aj #aj #abcd #cd #c #d #abcd ---- or #ab--- or #ab --- --- #ajj #ajj #aj #ajj #abcd #bcd #b #c #d #abcd ---- or #a---- or #a --- --- ---- #ajjj #ajjj #aj #ajj #ajjj #abcd #a #b #c #d #abcd ----- or --- --- ---- ----- #ajjjj #aj #ajj #ajjj #ajjjj #abcd #a #b #c #d #jabcd ------ or --- ---- ----- ------ #ajjjjj #ajj #ajjj #ajjjj #ajjjjj #abcd #a #b #c #d #jjabcd ------- or ---- ----- ------ ------- #ajjjjjj #ajjj #ajjjj #ajjjjj #ajjjjjj ,,ii4 #ajjc #a #c #jajjc is ------ or --- ------ #ajjjjj #ajj #ajjjjj #ajjc #a #c #ajjc is ----- or --- ----- #ajjjj #aj #ajjjj #ajjc #c #ajjc is ---- or #aj --- #ajj #ajj #ajjc #c #ajjc is ---- or #ajj --- #aj #aj ,,iii4 #a #b #h #c #abhc --- --- ---- ----- #aj #ajj #ajjj #ajjjj #a #jb #jjh #jjjc #a #jbhc #ab #jjhc #abh #jjjc #ajh #jbjc #ajjc #jbh #abjc #jjh #a is #ajjj in*es #b is #bjj #c is #cj #d is #d ,,iv4 ,9 #abcdefghi #e is _/ ( an in* in*es ! #f is _/ #g is _/ #h is _/ #i is _/ #acf4 ,! ciph]s on ! "r h& ( ! decimal po9t s]ve ! same purpose z ! ciph]s 9 7#aj74 ,!y are n c\nt$ z any ?+ !mselves1 b s]ve to %ew ! place 9 :i* ! accompany+ numb]s /&4 ,!y might be disp5s$ ) by writ+ ! numb]s 9 rul$ columns1 z 9 ! fir/ section4 ,!y are di/+ui%$ f ! numb]s :i* accompany !m by call+ ! latt] significant figures4 ,?us1 #jjjcgdg is a decimal ( sev5 places ) f\r significant figures1 #cdf is a decimal ( ?ree places ) ?ree significant figures1 c4 #acg4 ,! value ( a decimal is n alt]$ by putt+ any numb] ( ciph]s on its "r4 ,take1 = example1 #c & #cjj4 ,! fir/ 7#ace7 is _/1 & ! second _/1 :i* is made f ! fir/ by multiply+ bo? its num]ator & d5om9ator by #ajj1 & 7#ajh7 is ! same quantity4 #ach4 ,to r$uce two decimals to a common d5om9ator1 put z _m ciph]s on ! "r ( t :i* has ! small] numb] ( places z w make ! numb] ( places 9 bo? fractions ! same4 ,take1 = example1 #ed & #dcbig4 ,! fir/ is _/1 & ! second _/4 ,multiply ! num]ator & d5om9ator ( ! fir/ by #ajj 7#ajh71 :i* r$uces x to _/1 :i* has ! same d5om9ator z _/4 ,b _/ is #edjj 7#ace74 ,9 :ole numb]s1 ! decimal po9t %\ld be plac$ at ! end3 ?us1 #abi %\ld be writt5 #abi4 ,x is1 h[ev]1 usual to omit ! po9t2 b y mu/ recollect t #abi & #abijjj are ( ! same value1 s9ce ! fir/ is #abi & ! second _/4 #aci4 ,! rules :i* 7 giv5 9 ! la/ *apt] = addition1 subtraction1 multiplication1 & division1 apply to all fractions1 & !re=e to decimal fractions among ! re/4 ,b ! way ( writ+ decimal fractions1 :i* is expla9$ 9 ?is *apt]1 makes ! application ( ^! rules m simple4 ,we proce$ to ! diff]5t cases4 ,suppose x requir$ to add #dbfcd1 #debhjf1 #bjja1 & #ed4 ,by 7#aab7 ^! mu/ be r$uc$ to a common d5om9ator1 :i* is done 7#ach7 by writ+ !m z foll[s3 #dbfcdj1 #debhjf1 #bjjaj1 & #edjjjj4 ,^! are decimal fractions1 ^: num]ators are #dbfcdj1 #debhjf1 #bjjaj1 & #edjjjj1 & ^: common d5om9ator is #ajjjj4 ,by 7#aab7 _! sum is #dbfcdj #debhjf #bjjaj #edjjjj #adciaef --------------------------------1 :i* is ------- #ajjjj #ajjjj or #adciaef4 ,! simple/ way ( do+ ?is is z foll[s3 write ! decimals d[n "u "o ano!r1 s t ! decimal po9ts may fall "u "o ano!r1 ?us3 #dbfcd #debhjf #bjja #ed -------- #adciaef ,add ! diff]5t columns toge!r z 9 common addition1 & place ! decimal po9t "u ! o!r decimal po9ts4 ,,ex]cises4 ,:at are #aebg #fdgcbjid #bjjac #jjjjaigd2 #bbgfc #ajg #i #bfcagb #efgcbjja2 & #aaa #gg #jjci #jjadb #hhch8 ,answ]1 #aeicgccdacgd1 #eijcefbeb1 #ifiiab4 #adj4 ,suppose x requir$ to subtract #iajgcbd f #acgcba4 ,^! fractions :5 r$uc$ to a common d5om9ator are #iajgcbd & #acgcbajj 7#ach74 ,_! diff]5ce is !re=e #acgcbajj - #iajgcbd #dfbdggf ------------------1 :i* is ------- #ajjjjj #ajjjjj or #dfbdggf4 ,?is may be mo/ simply done z foll[s3 write ! less numb] "u ! great]1 s t its decimal po9t may fall "u t ( ! great]1 ?us3 #acgcba #iajgcbd --------- #dfbdggf ,subtract ! l[] f ! upp] l9e1 & :]ev] !re is a figure 9 "o l9e & n 9 ! o!r1 proce$ z if !re 7 a ciph] 9 ! vacant place4 ,,ex]cises4 ,:at is #abcfb - #bgdbbajg #e2 #iigfbjgcidb - #jjadcigfgbh2 & #ab #jc #jjd - #jjje8 ,answ]1 #abjhhbghic1 #iigfbjeieddcbgb2 & #abcce4 #ada4 ,! multiplication ( a decimal by #aj1 #ajj1 #ajjj1 c41 is p]=m$ by m]ely mov+ ! decimal po9t to ! "r4 ,suppose1 = example1 #acbjgi is to be multipli$ by #ajj4 ,! decimal is _/1 :i* multipli$ by #ajj is 7#aag7 _/1 or #acbjgi4 ,aga91 #acji #ajjjjj is _/ #ajjjjj1 or 7#aaf7 _/ or #acjijj4 ,f ^! & o!r in/ances we get ! foll[+ rule3 ,to multiply a decimal fraction by a decimal numb] 7#abf71 move ! decimal po9t z _m places to ! "r z !re are ciph]s 9 ! decimal numb]4 ,:5 ?is _c be done1 annex ciph]s to ! "r ( ! decimal 7#acg7 until x c4 #adb4 ,suppose x requir$ to multiply #agjcf by #dbg4 ,! fir/ ( ^! decimals is _/1 & ! second _/4 ,by 7#aah7 ! product ( ^! fractions has = its num]ator ! product ( #agjcf & #dbg1 & = its d5om9ator ! product ( #ajjj & #ajj2 !re=e ?is product is _/1 or #gbgdcgb4 ,?is may be done m %ortly by multiply+ ! two numb]s #agjcf & #dbg1 & cutt+ (f by ! decimal po9t z _m places z !re are decimal places bo? 9 #agjcf & #dbg1 because ! product ( two decimal numb]s w conta9 z _m ciph]s z !re are ciph]s 9 bo?4 #adc4 ,?is "q n[ arises3 ,:at if !re %\ld n be z _m figures 9 ! product z !re are decimal places 9 ! multipli] & multiplic& toge!r8 ,to see :at mu/ be done 9 ?is case1 multiply #agb by #aja1 or _/ by _/4 ,! product ( ^! two is _/1 or #jagcgb 7#ace74 ,!re=e1 :5 ! numb] ( places 9 ! product is n suffici5t to all[ ! rule ( ! la/ article to be foll[$1 z _m ciph]s mu/ be plac$ at ! beg9n+ z w make up ! defici5cy4 ,,additional ,,examples4 #jja #ja is #jjjja #ef #jjja is #jjef4 ,,ex]cises4 ,%ew t #cjjb #cjjb #c #c #b #c #jjb #jjb #jjb #aaefji #ecaia #hdd #hdd - #cabji #cabji #hbag #ajjja #h #aj #h #jja #aj #bag #jja #bag4 ,fraction4 ,squ>e4 ,cube4 #hbib #fhgegbfd #egjacebccjhh #jagc #jjjbiibi #jjjjjeagggag #adc #bjddi #bibdbjg #jji #jjjjha #jjjjjjgbi #aefbe #fd #ajjj #aefbe #fd #a #jaefbe #jjfd #jjja #aefbe #fd #a #aefbe #jfd #ajj #aefbejjj #jfd #ajjjjjj #add4 ,! division ( a decimal by a decimal numb]1 su* z #aj1 #ajj1 #ajjj1 c41 is p]=m$ by mov+ ! decimal po9t z _m places to ! left z !re are ciph]s 9 ! decimal numb]4 ,if !re are n places en\gh 9 ! divid5d to all[ ( ?is1 annex ciph]s to ! beg9n+ ( x until !re are4 ,= example1 divide #agcdbbi by #ajjj3 ! decimal fraction is _/1 :i* divid$ by #ajjj 7#abc7 is _/1 or #agcdbbi4 ,if1 9 ! same way1 #abajf be divid$ by #ajjjj1 ! result is #jjjabajf4 #ade4 ,be=e proce$+ to %ort5 ! rule = ! division ( "o decimal fraction by ano!r1 x w be necess>y to resume :at 0 said 9 7#abh7 ^u ! r$uction ( any fraction to a decimal fraction4 ,x 0 !re %ewn t _/ is ! same fraction z _/ or #dcge4 ,z ano!r example1 conv]t _/ into a decimal fraction4 ,foll[ ! same process z 9 7#abh71 ?us3 #abh7cjjjjjjjjjjj7bcdcge #bef ---- #ddj #chd ---- #efj #eab ---- #dhj #chd ---- #ifj #hif ---- #fdj #fdj --- #j ,s9ce #g ciph]s are us$1 x appe>s t #cjjjjjjj is ! fir/ ( ! s]ies #cj1 #cjj1 c41 :i* is divisible by #abh2 & !re=e _/ or1 :i* is ! same ?+ 7#ajh71 _/ is equal to _/ or #jbcdcge 7#ace74 ,f ^! examples ! rule = r$uc+ a fraction to a decimal is3 ,annex ciph]s to ! num]ator2 divide by ! d5om9ator1 & annex a ciph] to ea* rema9d] aft] ! figures ( ! num]ator are all us$1 proce$+ exactly z if ! num]ator _h an unlimit$ numb] ( ciph]s annex$ to x1 & 0 to be divid$ by ! d5om9ator4 ,cont9ue ?is process until !re is no rema9d]1 & obs]ve h[ _m ciph]s h be5 us$4 ,place ! decimal po9t 9 ! quoti5t s z to cut (f z _m figures z y h us$ ciph]s2 & if !re be n figures en\gh = ?is1 annex ciph]s to ! beg9n+ until !re are places en\gh4 #adf4 ,f :at 0 %ewn 9 7#abi71 x appe>s t x is n e fraction :i* c be r$uc$ to a decimal fraction4 ,x 0 !re %ewn1 h[ev]1 t !re is no fraction to :i* we may n f9d a decimal fraction z ne> z we please4 ,?us1 _/1 _/1 _/1 _/1 _/1 c41 or #a1 #ad1 #adb1 #adbh1 #adbhe1 7 %ewn to be fractions :i* approa* ne>] & ne>] to _/4 ,to f9d ei!r ( ^! fractions1 ! rule is ! same z t 9 ! la/ article1 ) ?is exception1 t1 ,i4 in/ead ( /opp+ :5 !re is no rema9d]1 :i* nev] happ5s1 /op at any "p ( ! process1 & make z _m decimal places 9 ! quoti5t z are equal 9 numb] to ! numb] ( ciph]s :i* h be5 us$1 annex+ ciph]s to ! beg9n+ :5 ?is _c be done1 z be=e4 ,,ii4 ,in/ead ( obta9+ a fraction :i* is exactly equal to ! fraction f :i* we set out1 we get a fraction :i* is v ne> to x1 & may get "o /ill ne>]1 by us+ m ( ! quoti5t4 ,?us1 #adbh is v ne> to _/1 b n s ne> z #adbheg2 nor is ?is la/1 9 its turn1 s ne> z #adbhegadbheg1 c4 #adg4 ,if !re %\ld be ciph]s 9 ! num]ator ( a fraction1 ^! mu/ n be reckon$ ) ! numb] ( ciph]s :i* are necess>y 9 ord] to foll[ ! rule = *ang+ x into a decimal fraction4 ,take1 = example1 _/2 annex ciph]s to ! num]ator1 & divide by ! d5om9ator4 ,x appe>s t #ajjj is divisible by #abe1 & t ! quoti5t is #h4 ,"o ciph] only has be5 annex$ to ! num]ator1 & !re=e #ajj divid$ by #abe is #h4 ,_h ! fraction be5 _/1 s9ce #ajjj divid$ by #abe gives #h1 & ?ree ciph]s w\ld h be5 annex$ to ! num]ator1 ! fraction w\ld h be5 #jjh4 #adh4 ,suppose t ! giv5 fraction has ciph]s at ! "r ( its d5om9ator2 = example1 _/4 ,!n annex+ a ciph] to ! num]ator is ! same ?+ z tak+ "o away f ! d5om9ator2 =1 7#ajh7 _/ is ! same ?+ z _/1 & _/ z _/4 ,! rule1 !re=e1 is 9 ?is case3 ,take away ! ciph]s f ! d5om9ator4 ,,ex]cises4 ,r$uce ! foll[+ fractions to decimal fractions3 #a #cf #big #a ---1 ----1 ----1 & ---4 #hjj #abej #fd #abh ,answ]1 #jjabe1 #jbhh1 #dfdjfbe1 & #jjghabe4 ,f9d decimals ( #f places v ne> to ! foll[+ fractions3 #bg #aef #bb #aid #bfcg #a #a #c --1 ---1 -----1 ---1 ----1 ----1 ---1 & ---4 #di #cc #cgjjj #ac #iijg #bijh #dff #bgg ,answ]1 #eeajbj1 #dgbgbgb1 #jjjeid1 #adibcjgf1 #bffage1 #jjjcdc1 #jjbade1 & #jajhcj4 #adi4 ,f 7#aba7 x appe>s1 t if two fractions h ! same d5om9ator1 ! fir/ may be divid$ by ! second by divid+ ! num]ator ( ! fir/ by ! num]ator ( ! second4 ,suppose x requir$ to divide #aggfb by #fbe4 ,^! fractions 7#ach71 :5 r$uc$ to a common d5om9ator1 are #aggfb & #fbej1 or _/ & _/4 ,_! quoti5t is !re=e _/1 :i* mu/ n[ be r$uc$ to a decimal fraction by ! la/ rule4 ,! process at full l5g? is z foll[s3 ,leave out ! ciph] 9 ! d5om9ator1 & annex ciph]s to ! num]ator1 or1 :i* w d z well1 to ! rema9d]s1 :5 x becomes necess>y1 & divide z 9 7#ade74 #fbe7aggfb7bhdaib #abej ----- #ebfb #ejjj ----- #bfbj #bejj ----- #abjj #fbe ----- #egej #efbe ----- #abej #abej ---- #j ,"h f\r ciph]s h be5 annex$ to ! num]ator1 & "o has be5 tak5 f ! d5om9ator4 ,make five decimal places 9 ! quoti5t1 :i* !n becomes #bhdaib1 & ?is is ! quoti5t ( #aggfb divid$ by #fbe4 #aej4 ,! rule = division ( "o decimal by ano!r is z foll[s3 ,equalise ! numb] ( decimal places 9 ! divid5d & divisor1 by annex+ ciph]s to t :i* has fewe/ places4 ,!n1 fur!r1 annex z _m ciph]s to ! divid5d#ah z x is requir$ to h decimal places1 ?r[ away ! decimal po9t1 & op]ate z 9 common division4 ,make ! requir$ numb] ( decimal places 9 ! quoti5t4 #ah ,or remove ciph]s f ! divisor2 or make up ! numb] ( ciph]s p>tly by remov+ f ! divisor & annex+ to ! divid5d1 if !re be n a suffici5t numb] 9 ! divisor4 ,?us1 to divide #fgagc by #jad to ?ree decimal places1 ,i fir/ write #fgagc & #jadj1 ) f\r places 9 ea*4 ,hav+ to provide = ?ree decimal places1 ,i %\ld annex ?ree ciph]s to #fgagc2 b1 obs]v+ t ! divisor #jadj has "o ciph]1 ,i /rike t "o out & annex two ciph]s to #fgagc4 ,?r[+ away ! decimal po9ts1 !n divide #fgagcjj by #jad or #ad 9 ! usual way1 :i* gives ! quoti5t #dgihjg & ! rema9d] #b4 ,h5ce #dgihjg is ! answ]4 ,! common rule is3 ,let ! quoti5t conta9 z _m decimal places z !re are decimal places 9 ! divid5d m ?an 9 ! divisor4 ,b ?is rule becomes inop]ative except :5 !re are m decimals 9 ! divid5d ?an 9 ! divisor1 & a numb] ( ciph]s mu/ be annex$ to ! =m]4 ,! rule 9 ! text am\nts to ! same ?+1 & provides = an assign$ numb] ( decimal places4 ,b ! /ud5t is recomm5d$ to make himself famili> ) ! rule ( ! *>act]i/ic giv5 9 ! ,app5dix1 & also to accu/om himself to reason out ! place ( ! decimal po9t4 ,?us1 x %\ld be visible1 t #bfaai #gbdcf has "o figure be=e ! decimal po9t1 & t #bfaai #gbdcf has "o ciph] aft] x1 prec$+ all significant figures4 ,or ! foll[+ rule may be us$3 ,expunge ! decimal po9t ( ! divisor1 & move t ( ! divid5d z _m places to ! "r z !re 7 places 9 ! divisor1 us+ ciph]s if necess>y4 ,!n proce$ z 9 common division1 mak+ "o decimal place 9 ! quoti5t = e decimal place ( ! f9al divid5d :i* is us$4 ,?us #agcad divid$ by #fab is #agcad divid$ by #fab1 & ! decimal po9t mu/ prec$e ! fir/ figure ( ! quoti5t4 ,b #agcad divid$ by #ffage is #agcad by #ffage2 & s9ce ?ree decimal places ( #agcadjjj 444 mu/ be us$ be=e a quoti5t figure c be f\nd1 t quoti5t figure is ! ?ird decimal place1 or ! quoti5t is #jjb44444 ,,examples4 #ca #jjjfb ----- #abdj1 ------ #jjjifhge #jjbe #fd ,,ex]cises4 #aejjf #aejjf - #jjd #jjd ,%ew t ----------------------------- #aejjb1 #aeja & t #ja #ja #ja #bi #bi #bi --------------------------------- #bi #bi - #bi #ja #ja #ja #bia #a #a #cfe ,:at are -------1 ---------1 & ------1 z f> z #f places #cadaei #bgahbhah #ahcdi ( decimals8--,answ]1 #cahcaj1 #cfghgi1 & #aihibjibba4 ,calculate #aj t]ms ( ea* ( ! foll[+ s]ies1 z f> z #e places ( decimals4 #a #a #a #a #a --- ----- --------- ------------- c4 #agahbd4 #b #b #c #b #c #d #b #c #d #e #a #a #a #a #a --- --- --- --- c4 #bibhie4 #b #c #d #e #hj #ha #hb #hc #hd ---- ---- ---- ---- ---- c4 #ihhbhf4 #ha #hb #hc #hd #he #aea4 ,we n[ ent] ^u me?ods by :i* unnecess>y tr\ble is sav$ 9 ! computation ( decimal quantities4 ,& fir/1 suppose a numb] ( miles has be5 measur$1 & f\nd to be #aghdfbag miles4 ,if y 7 ask$ h[ _m miles !re are 9 ?is di/ance1 & a r\gh answ] 7 requir$ :i* %\ld give miles only1 & n p>ts ( miles1 y w\ld probably say #ag4 ,b ?is1 ?\gh ! numb] ( :ole miles conta9$ 9 ! di/ance1 is n ! ne>e/ numb] ( miles2 =1 s9ce ! di/ance is m ?an #ag miles & #h t5?s1 & !re=e m ?an #ag miles & a half1 x is ne>] ! tru? to say1 x is #ah miles4 ,?is1 ?\gh too great1 is n s mu* too great z ! o!r 0 too little1 & ! error is n s great z half a mile4 ,aga91 if ! same 7 requir$ )9 a t5? ( a mile1 ! correct answ] is #agh2 = ?\gh ?is is too little by #jdfbag1 yet x is n s mu* too little z #agi is too great2 & ! error is less ?an half a t5?1 or _/4 ,aga91 ! same di/ance1 )9 a hundr$? ( a mile1 is m correctly #aghe ?an #aghd1 s9ce ! la/ is too little by #jjfbag1 :i* is great] ?an ! half ( #ja2 & !re=e #aghd #ja is ne>] ! tru? ?an #aghd4 ,h5ce ?is g5]al rule3 ,:5 a c]ta9 numb] ( ! decimals giv5 is suffici5tly accurate = ! purpose1 /rike (f ! re/ f ! "r h&1 obs]v+1 if ! fir/ figure /ruck (f be equal to or great] ?an #e1 to increase ! la/ rema9+ figure by #a4 ,! foll[+ are examples ( a decimal abbreviat$ by "o place at a "t4 #cadaei1 #cadaf1 #cadb1 #cad1 #ca1 #cj #bgahbhah1 #bgahbhb1 #bgahbh1 #bgahc1 #bgah1 #bgb1 #bg1 #cj #aiiai1 #aiib1 #aii1 #bjj1 #bj #aeb4 ,9 multiplication & division x is useless to reta9 m places ( decimals 9 ! result ?an 7 c]ta9ly correct 9 ! multipli]1 c41 :i* gave t result4 ,suppose1 = example1 t #iih & #hif are di/ances 9 in*es :i* h be5 measur$ correctly to two places ( decimals1 t is1 )9 half a hundr$? ( an in* ea* way4 ,! real value ( t :i* we call #iih may be any :]e betwe5 #iige & #iihe1 & t ( #hif may be any :]e betwe5 #hiee & #hife4 ,! product1 !re=e1 ( ! numb]s :i* repres5t ! correct di/ances w lie betwe5 #iige #hiee & #iihe #hife1 t is1 tak+ ?ree decimal places 9 ! products1 betwe5 #hicbf & #hieaf4 ,! product ( ! actual numb]s giv5 is #hidbjh4 ,x appe>s1 !n1 t 9 ?is case no m ?an ! :ole numb] #hi c be dep5d$ ^u 9 ! product1 or1 at mo/1 ! fir/ place ( decimals4 ,! reason is1 t ! error made 9 measur+ #hif1 ?\gh only 9 ! ?ird place ( decimals1 is 9 ! multiplication increas$ at lea/ #iige1 or ne>ly #aj times2 & !re=e affects ! second place4 ,! foll[+ simple rule w enable u to judge h[ f> a product is to be dep5d$ ^u4 ,let a be ! multipli]1 & b ! multiplic&2 if ^! be true only to ! fir/ decimal place1 ! product is )9 7a b7_/#bjai ( ! tru?2 if to two decimal places1 )9 7a b7_/#bjj2 if to ?ree1 )9 7a b7_/#bjjj2 & s on4 ,?us1 9 ! above example1 we h #iih & #hif1 :i* are true to two decimal places3 _! sum divid$ by #bjj is #jidg1 & _! product is #hidbjh1 :i* is !re=e )9 #jidg ( ! tru?4 ,if1 9 fact1 we increase & dim9i% #hidbjh by #jidg1 we get #hieaee & #hicbfa1 :i* are v ne>ly ! limits f\nd )9 :i* ! product mu/ lie4 ,we see1 !n1 t we _c 9 ?is case dep5d ^u ! fir/ place ( decimals1 z 7#aea7 an error ( #je _c exi/ if ?is place be correct2 & "h is a possible error ( #ji & upw>ds4 ,x is h>dly necess>y to say1 t if ! numb]s giv5 be exact1 _! product is exact also1 & t ?is article applies :]e ! numb]s giv5 are correct only to a c]ta9 numb] ( decimal places4 ,! rule is3 ,take half ! sum ( ! multipli] & multiplic&1 remove ! decimal po9t z _m places to ! left z !re are correct places ( decimals 9 ei!r ! multipli] or multiplic&2 ! result is ! quantity )9 :i* ! product c be dep5d$ ^u4 ,9 division1 ! rule is3 ,proce$ z 9 ! la/ rule1 putt+ ! divid5d & divisor 9 place ( ! multipli] & multiplic&1 & divide by ! squ>e ( ! divisor2 ! quoti5t w be ! quantity )9 :i* ! division ( ! fir/ divid5d & divisor may be dep5d$ ^u4 ,?us1 if #agcbd be divid$ by #echji1 bo? be+ correct to ! ?ird place1 _! half sum w be #ceeff1 :i*1 by ! la/ rule1 is made #jceeff1 & is to be divid$ by ! squ>e ( #echji1 or1 :i* w d z well = our purpose1 ! squ>e ( #ej1 or #bejj4 ,! result is some?+ less ?an #jjjjb1 s t ! quoti5t ( #agcbd & #echji c be dep5d$ on to f\r places ( decimals4 #ai ,^! are n q correct1 b suffici5tly s = e practical purpose4 #aec4 ,x is requir$ to multiply two decimal fractions toge!r1 s z to reta9 9 ! product only a giv5 numb] ( decimal places1 & disp5se ) ! tr\ble ( f9d+ ! re/4 ,fir/1 x is evid5t t we may write ! figures ( any multipli] 9 a contr>y ord] 7= example1 #dcba in/ead ( #abcd71 provid$ t 9 ! op]ation we move ea* l9e "o place to ! "r in/ead ( to ! left1 z 9 ! foll[+ example3 #bbba #bbba #abcd #dcba ---- ---- #hhhd #bbba #fffc #dddb #dddb #fffc #bbba #hhhd ------- ------- #bgdjgad #bgdjgad ,suppose n[ we wi% to multiply #cdhhdad by #eacjgdb1 res]v+ only f\r decimal places 9 ! product4 ,if we rev]se ! multipli]1 & proce$ 9 ! mann] j po9t$ out1 we h ! foll[+3 #cdhhdad #bdgjcae --------- #agddbjgj #cdhhdad #ajdfebdb #bddahhih #aciecfef #figfhbh ---------------- #aghihaebbbcahh ,cut (f1 by a v]tical l9e1 ! fir/ f\r places ( decimals1 & ! columns :i* produc$ !m4 ,x is pla9 t 9 =m+ our abbreviat$ rule1 we h to consid] only1 ,i4 all t is on ! left ( ! v]tical l9e2 ,,ii4 all t is c>ri$ f ! fir/ column on ! "r ( ! l9e4 ,on look+ at ! fir/ column to ! left ( ! l9e1 we see #d1 #d1 #h1 #e1 #i1 ( :i* ! fir/ #d comes f #d #a1bj ! second #d f #a #c1 ! #h f #h #g1 ! #e f #h #d1 & ! #i f #d #b4 ,if1 !n1 we arrange ! multiplic& & ! rev]s$ multipli] ?us1 #cdhhdad #bdgjcae ea* figure ( ! multipli] is plac$ "u ! fir/ figure ( ! multiplic& :i* is us$ ) x 9 =m+ ! fir/ f\r places ( decimals4 ,& "h obs]ve1 t ! units' figure 9 ! multipli] #eacjgdb1 viz4 #a1 comes "u #d1 ! f\r? decimal place 9 ! multiplic&4 ,if !re _h be5 no c>ry+ f ! "r ( ! v]tical l9e1 ! rule w\ld h be53 ,rev]se ! multipli]1 & place x "u ! multiplic&1 s t ! figure :i* 0 ! units' figure 9 ! multipli] may /& "u ! la/ place ( decimals 9 ! multiplic& :i* is to be pres]v$2 place ciph]s ov] ^? figures ( ! multipli] :i* h none ( ! multiplic& above !m1 if !re be any3 proce$ to multiply 9 ! usual way1 b beg9 ea* figure ( ! multipli] ) ! figure ( ! multiplic& :i* comes above x1 tak+ no acc\nt ( ^? on ! "r3 place ! fir/ figures ( all ! l9es "u "o ano!r4 ,to correct ?is rule1 s z to all[ = :at is c>ri$ f ! "r ( ! v]tical l9e1 obs]ve t ?is consi/s ( two p>ts1 #a/1 :at is c>ri$ directly 9 ! =mation ( ! diff]5t l9es1 & #b;dly1 :at is c>ri$ f ! addition ( ! fir/ column on ! "r4 ,! fir/ ( ^! may be tak5 into acc\nt by beg9n+ ea* figure ( ! multipli] ) ! "o :i* comes on its "r 9 ! multiplic&1 & c>ry+ ! t5s to ! next figure z usual1 b )\t writ+ d[n ! units4 ,b bo? may be all[$ = at once1 ) suffici5t correctness1 on ! pr9ciple ( 7#aea71 by c>ry+ #a f #e up to #ae1 #b f #ae up to #be1 c42 t is1 by c>ry+ ! ne>e/ t54 ,?us1 = #cg1 #d w\ld be c>ri$1 #cg be+ ne>] to #dj ?an to #cj4 ,?is w n always give ! la/ place q correctly1 b ! error may be avoid$ by sett+ out s z to keep "o m place ( decimals 9 ! product ?an is absolutely requir$ to be correct4 ,! rule1 !n1 is z foll[s3 #bj ,! #a "h means t ! #a is 9 ! multipli]4 #aed4 ,to multiply two decimals toge!r1 reta9+ only n decimal places4 ,i4 ,rev]se ! multipli]1 /rike out ! decimal po9ts1 & place ! multipli] "u ! multiplic&1 s t :at 0 its units' figure %all fall "u ! n decimal place ( ! multiplic&1 plac+ ciph]s1 if necess>y1 s t e place ( ! multipli] %all h a figure or ciph] above x4 ,,ii4 ,proce$ to multiply z usual1 beg9n+ ea* figure ( ! multipli] ) ! "o :i* is 9 ! place to its "r 9 ! multiplic&3 d n set d[n ?is fir/ figure1 b c>ry its ne>e/ t5 to ! next1 & proce$4 ,,iii4 ,place ! fir/ figures ( all ! l9es "u "o ano!r2 add z usual2 & m>k (f n places f ! "r = decimals4 ,x is requir$ to multiply #acfdjgb by #acjfji1 reta9+ #g decimal places4 #acfdjgbjjj #ijfjca ---------- #acfdjgbjjj #djibbafjj #hahddcb #abbgff ----------- #aghafjjgih ,9 ! foll[+ examples ! fir/ two l9es are ! multiplic& & multipli]2 & ! numb] ( decimals to be reta9$ w be se5 f ! results4 #ddgafah #ccaffbdh #cdfdajaf #cggaibad #adadbacf #agcbejh #cggaibad #jccaffbdh #cdfdajafj #hafagdd #fcabdada #hjebcga -------- ---------- ------------ #aejhgfhf #ccaffbe #cdfdajafj #aejhgfh #acbffej #bdbdhgaab #bfdjcd #ccaff #ajcibcje #cggb #acbff #fibhbj #bbfc #ffc #agcbje #ch #cc #bgga #cj #aj ------------ --------- #b #fjjaehcgc #afhffeia -------- #dfijdae ,ex]cises may be got f article 7#adc74 #aee4 ,) reg>d to division1 take any two numb]s1 = example1 #afhjdcgiba & #cadb1 & divide ! fir/ by ! second1 z f> z any requir$ numb] ( decimal places1 = example1 five4 ,?is gives ! foll[+3 #cadb7afhjdcgiba7ecdhcj #aegaj ------- #ajidc #idbf ----- #aeagg 7,a7 #abefh ---- ------ #bfji #bfjii #bead #beacf ---- ------ #ie #ifcb #id #idbf -- ----- #a #bjfa ,n[ cut (f by a v]tical l9e1 z 9 7#aec71 all ! figures :i* come on ! "r ( ! fir/ figure #b1 9 ! la/ rema9d] #bjfa4 ,z 9 multiplication1 we may obta9 all t is on ! left ( ! v]tical l9e by an abbreviat$ me?od1 z repres5t$ at 7,a74 ,aft] :at has be5 said on multiplication1 x is useless to g fur!r into ! detail2 ! foll[+ rule w be suffici5t3 ,to divide "o decimal by ano!r1 reta9+ only n places3 ,proce$ "o /ep 9 ! ord9>y division1 & det]m9e1 by 7#aej71 9 :at place is ! quoti5t s obta9$2 proce$ 9 ! ord9>y way1 until ! numb] ( figures rema9+ to be f\nd 9 ! quoti5t is less ?an ! numb] ( figures 9 ! divisor3 if ?is %\ld be already ! case1 proce$ no fur!r 9 ! ord9>y way4 ,in/ead ( annex+ a figure or ciph] to ! rema9d]1 cut (f a figure f ! divisor1 & proce$ "o /ep ) ?is curtail$ divisor z usual1 rememb]+1 h[ev]1 9 multiply+ ?is divisor1 to c>ry ! ne>e/ t51 z 9 7#aed71 f ! figure :i* 0 /ruck (f2 repeat ?is1 /rik+ (f ano!r figure ( ! divisor1 & s on1 until no figures are left4 ,s9ce we "k f ! beg9n+ 9 :at place ! fir/ figure ( ! quoti5t is1 & also h[ _m decimals are requir$1 we c tell f ! beg9n+ h[ _m figures !re w be 9 ! :ole quoti5t4 ,if ! divisor conta9 m figures ?an ! quoti5t1 x w be unnecess>y to use !m3 & !y may be reject$1 ! re/ be+ correct$ z 9 7#aea73 if !re be ciph]s at ! beg9n+ ( ! divisor1 if x be1 = example1 #cagh #jjcagh1 s9ce ?is is -----1 #ajj divide by #cagh 9 ! usual way1 & aft]w>ds multiply ! quoti5t by #ajj1 or remove ! decimal po9t two places to ! "r4 ,if1 !re=e1 six decimals be requir$1 eight places mu/ be tak5 9 divid+ by #cagh1 = an obvi\s reason4 ,9 f9d+ ! la/ figure ( ! quoti5t1 ! ne>e/ %\ld be tak51 z 9 ! second ( ! subjo9$ examples4 ,places requir$1 #b #h ,divisor1 #dadcb #cadaeibg ,divid5d1 #fgcadhi #bgahbhahj #dadcb #beacbgdaf -------- ---------- #behhbh #bjejjgfd #bdheib #ahhdieef ------- -------- #ajbcgba #afeabjh #hbhf #aegjgif ----- ------- #aiea #hjdab #afeg #fbhcb ----- ----- #bid #agehj #bij #aegjh --- ----- #d #ahgb #d #aega - ---- #j #cja #bhc --- #ah #ai -- ,quoti5t1 #afbdga #hfebeeif #ba ,?is is writt5 #g in/ead ( #f1 because ! figure :i* is ab&on$ 9 ! divid5d is #i 7#aea74 ,examples may be obta9$ f 7#adc7 & 7#aej74 ,,section ,,vii4 ,,on ,,! ,,extraction ,,( ,,! ,,squ>e ,,root4 #aef4 ,we h already rem>k$ 7#ff71 t a numb] multipli$ by itself produces :at is call$ ! squ>e ( t numb]4 ,?us1 #afi1 or #ac #ac1 is ! squ>e ( #ac4 ,conv]sely1 #ac is call$ ! squ>e root ( #afi1 & #e is ! squ>e root ( #be2 & any numb] is ! squ>e root ( ano!r1 :i* :5 multipli$ by itself w produce t o!r4 ,! squ>e root is signifi$ by ! sign or 2 ?us1 #be means ! squ>e root ( #be1 or #e2 7#af #i7 means ! squ>e root ( #af #i1 & is #e1 & mu/ n be conf\nd$ ) #af #i1 :i* is #d #c1 or #g4 #aeg4 ,! foll[+ equations are evid5t f ! def9ition3 a a a aa a ab ab ab 7a b7 7a b7 a a b b ab :5ce a b ab #aeh4 ,x does n foll[ t a numb] has a squ>e root because x has a squ>e2 ?us1 ?\gh #e c be multipli$ by itself1 !re is no numb] :i* multipli$ by itself w produce #e4 ,x is prov$ 9 algebra1 t no fraction#bb multipli$ by itself c produce a :ole numb]1 :i* may be f\nd true 9 any numb] ( in/ances2 !re=e #e has nei!r a :ole nor a fractional squ>e root2 t is1 x has no squ>e root at all4 ,nev]!less1 !re are me?ods ( f9d+ fractions ^: squ>es %all be z ne> to #e z we please1 ?\gh n exactly equal to x4 ,"o ( ^! me?ods gives _/1 ^: squ>e1 viz4 #aeabg #aeabg #bbhhbfabi ----- ----- or ---------1 #fgfe #fgfe #degfebbe diff]s f #e by only _/1 :i* is less ?an #jjjjjja3 h5ce we are enabl$ to use #e 9 ari?metical & algebraical reason+3 b :5 we come to ! practice ( any problem1 we mu/ sub/itute = #e "o ( ! fractions ^: squ>e is ne>ly #e1 & on ! degree ( accuracy we want1 dep5ds :at fraction is to be us$4 ,= "s purposes1 _/ may be suffici5t1 z its squ>e only diff]s f #e by _/2 = o!rs1 ! fraction fir/ giv5 might be necess>y1 or "o ^: squ>e is ev5 ne>] to #e4 ,we proce$ to %ew h[ to f9d ! squ>e root ( a numb]1 :5 x has "o1 & f !nce h[ to f9d fractions ^: squ>es %all be z ne> z we please to ! numb]1 :5 x has n4 ,we premise1 :at is suffici5tly evid5t1 t ( two numb]s1 ! great] has ! great] squ>e2 & t if "o numb] lie betwe5 two o!rs1 its squ>e lies betwe5 ! squ>es ( ^? o!rs4 #bb ,mean+1 ( c\rse1 a really fractional numb]1 su* z or _/1 n "o :i*1 ?\gh fractional 9 =m1 is :ole 9 reality1 su* z _/ or _/4 #aei4 ,let x be a numb] consi/+ ( any numb] ( p>ts1 = example1 f\r1 viz4 a1 b1 c1 & d2 t is1 let x a b c d ,! squ>e ( ?is numb]1 f\nd z 9 7#fh71 w be aa #b;a7b c d7 bb #b;b7c d7 cc #b;cd dd ,! rule !re f\nd = squ>+ a numb] consi/+ ( p>ts 03 ,squ>e ea* "p1 & multiply all t come aft] by twice t "p1 ! sum ( all ! results s obta9$ w be ! squ>e ( ! :ole numb]4 ,9 ! expression above obta9$1 in/ead ( multiply+ #b;a by ea* ( ! succe$+ p>ts1 b1 c1 & d1 & add+ ! results1 we multipli$ #b;a by ! sum ( all ! succe$+ p>ts1 :i* 7#eb7 is ! same ?+2 & z ! p>ts1 h[ev] dispos$1 make up ! numb]1 we may rev]se _! ord]1 putt+ ! la/ fir/1 c42 & ! rule = squ>+ w be3 ,squ>e ea* "p1 & multiply all t come be=e by twice t "p4 ,h5ce a rev]se rule = extract+ ! squ>e root pres5ts itself ) m ?an usual simplicity4 ,x is3 ,to extract ! squ>e root ( a numb] ,n1 *oose a numb] ,a1 & see if ,n w be> ! subtraction ( ! squ>e ( ,a2 if s1 take ! rema9d]1 *oose a second numb] ,b1 & see if ! rema9d] w be> ! subtraction ( ! squ>e ( ,b1 & twice ,b multipli$ by ! prec$+ "p ,a3 if x w1 !re is a second rema9d]4 ,*oose a ?ird numb] ,c1 & see if ! second rema9d] w be> ! subtraction ( ! squ>e ( ,c1 & twice ,c multipli$ by ,a ,b3 g on 9 ?is way ei!r until !re is no rema9d]1 or else until ! rema9d] w n be> ! subtraction aris+ f any new "p1 ev5 ?\gh t "p 7 ! lea/ numb]1 :i* is #a4 ,9 ! fir/ case1 ! squ>e root is ! sum ( ,a1 ,b1 ,c1 c42 9 ! second1 !re is no squ>e root4 #afj4 ,= example1 ,i wi% to "k if #bjbe has a squ>e root4 ,i *oose #bj z ! fir/ "p1 & f9d t #djj1 ! squ>e ( #bj1 subtract$ f #bjbe1 gives #afbe1 ! fir/ rema9d]4 ,i aga9 *oose #bj1 ^: squ>e1 toge!r ) twice itself1 multipli$ by ! prec$+ "p1 is #bj #bj #b #bj #bj1 or #abjj2 :i* subtract$ f #afbe1 ! fir/ rema9d]1 gives #dbe1 ! second rema9d]4 ,i *oose #g = ! ?ird "p1 :i* appe>s to be too great1 s9ce #g #g1 increas$ by #b #g multipli$ by ! sum ( ! prec$+ p>ts #bj #bj1 gives #fji1 :i* is m ?an #dbe4 ,i !re=e *oose #e1 :i* closes ! process1 s9ce #e #e1 toge!r ) #b #e multipli$ by #bj #bj1 gives exactly #dbe4 ,! squ>e root ( #bjbe is !re=e #bj #bj #e1 or #de1 :i* w be f\nd1 by trial1 to be correct2 s9ce #de #de #bjbe4 ,aga91 ,i ask if #accdj has1 or has n1 a squ>e root4 ,let #ajj be ! fir/ "p1 ^: squ>e is #ajjjj1 & ! fir/ rema9d] is #ccdj4 ,let #aj be ! second "p4 ,"h #aj #aj #b #aj #ajj is #bajj1 & ! second rema9d]1 or #ccdj-bajj1 is #abdj4 ,let #e be ! ?ird "p2 !n #e #e #b #e 7#ajj #aj7 is #aabe1 :i*1 subtract$ f #abdj1 leaves #aae4 ,!re is1 !n1 no squ>e root2 = a s+le additional unit w give a subtraction ( #a #a #b #a 7#ajj #aj #e71 or #bca1 :i* is great] ?an #aae4 ,b if ! numb] propos$ _h be5 less by #aae1 ea* ( ! rema9d]s w\ld h be5 #aae less1 & ! la/ rema9d] w\ld h be5 no?+4 ,!re=e #accdj-aae1 or #acbbe1 has ! squ>e root #ajj #aj #e1 or #aae2 & ! answ] is1 t #accdj has no squ>e root1 & t #acbbe is ! next numb] bel[ x :i* has "o1 namely1 #aae4 #afa4 ,x only rema9s to put ! rule 9 su* a %ape z w guide u to ^? p>ts :i* x is mo/ conv5i5t to *oose4 ,x is evid5t 7#eg7 t any numb] :i* t]m9ates ) ciph]s1 z #djjj1 has d\ble ! numb] ( ciph]s 9 its squ>e4 ,?us1 #djjj #djjj #afjjjjjj2 !re=e1 any squ>e numb]1#bc z #di1 ) an ev5 numb] ( ciph]s annex$1 z #dijjjj1 is a squ>e numb]4 ,! root#bd ( #dijjjj is #gjj4 ,?is be+ premis$1 take any numb]1 = example1 #gfagf2 sett+ out f ! "r h& t[>ds ! left1 cut (f two figures2 !n two m1 & s on1 until "o or two figures only are left3 ?us1 #g1fa1gf4 ,?is numb] is great] ?an #g1jj1jj1 ( :i* ! fir/ figure is n a squ>e numb]1 ! ne>e/ squ>e bel[ x be+ #d4 ,h5ce1 #d1jj1jj is ! ne>e/ squ>e numb] bel[ #g1jj1jj1 :i* has f\r ciph]s1 & its squ>e root is #bjj4 ,let ?is be ! fir/ "p *os53 its squ>e subtract$ f #gfagf leaves #cfagf1 ! fir/ rema9d]2 & x is evid5t t we h obta9$ ! highe/ numb] ( ! highe/ d5om9ation :i* is to be f\nd 9 ! squ>e root ( #gfagf2 = #cjj is too great1 its squ>e1 #i1jj1jj1 be+ great] ?an #gfagf3 & any d5om9ation high] ?an hundr$s has a squ>e /ill great]4 ,x rema9s1 !n1 to *oose a second "p1 z 9 ! examples ( 7#afj71 ) ! rema9d] #cfagf4 ,?is "p _c be z great z #ajj1 by :at has j be5 said2 its highe/ d5om9ation is !re=e a numb] ( t5s4 ,let ,n /& = a numb] ( t5s1 :i* is "o ( ! simple numb]s #a1 #b1 #c1 c42 t is1 let ! new "p be #aj,n1 ^: squ>e is #aj,n #aj,n1 or #ajj,,nn1 & ^: d\ble multipli$ by ! =m] "p is #bj,n #bjj1 or #djjj,n2 ! two toge!r are #djjj,n #ajj,,nn4 ,n[1 ,n mu/ be s tak5 t ?is may n be great] ?an #cfagf3 /ill m #djjj,n mu/ n be great] ?an #cfagf4 ,we may !re=e try1 = ,n1 ! numb] ( times :i* #cfagf conta9s #djjj1 or t :i* #cf conta9s #d4 ,! rem>k 9 7#hj7 applies "h4 ,let u try #i t5s or #ij4 ,!n1 #b #ij #bjj #ij #ij1 or #ddajj1 is to be subtract$1 :i* is too great1 s9ce ! :ole rema9d] is #cfagf4 ,we !n try #h t5s or #hj1 :i* gives #b #hj #bjj #hj #hj1 or #chdjj1 :i* is likewise too great4 ,on try+ #g t5s1 or #gj1 we f9d #b #gj #bjj #gj #gj1 or #cbijj1 :i* subtract$ f #cfagf gives #cbgf1 ! second rema9d]4 ,! re/ ( ! squ>e root c only be units4 ,z be=e1 let ,n be ?is numb] ( units4 ,!n1 ! sum ( ! prec$+ p>ts be+ #bjj #gj1 or #bgj1 ! numb] to be subtract$ is #bgj #b,n ,,nn1 or #edj,n ,,nn4 ,h5ce1 z be=e1 #edj,n mu/ be less ?an #cbgf1 or ,n mu/ n be great] ?an ! numb] ( times :i* #cbgf conta9s #edj1 or 7#hj7 :i* #cbg conta9s #ed4 ,we !re=e try if #f w d1 :i* gives #b #f #bgj #f #f1 or #cbgf1 to be subtract$4 ,?is be+ exactly ! second rema9d]1 ! ?ird rema9d] is no?+1 & ! process is f9i%$4 ,! squ>e root requir$ is !re=e #bjj #gj #f1 or #bgf4 #bc ,by squ>e numb] ,i mean1 a numb] :i* has a squ>e root4 ,?us1 #be is a squ>e numb]1 b #bf is n4 #bd ,! t]m #8root' is frequ5tly us$ z an abbreviation ( squ>e root4 ,! process ( =m+ ! numb]s to be subtract$ may be %ort5$ ?us4 ,let ,a be ! sum ( ! p>ts already f\nd1 & ,n a new "p3 !re mu/ !n be subtract$ #b;,,an ,,nn1 or 7#ed7 #b;,a ,n multipli$ by ,n4 ,! rule1 !re=e1 = =m+ x is3 ,d\ble ! sum ( all ! prec$+ p>ts1 add ! new "p1 & multiply ! result by ! new "p4 #afb4 ,! process ( ! la/ article is z foll[s3 #g1fa1gf7bjj #g1fa1gf7bgf #d #jj #jj #gj #d ------- #f --- #djj7c1fa1gf #dg7cfa #gj7c #bi #jj #cbi ------- ----- #djj7 #cb #gf #edf7cbgf #adj7 #cb #gf #cbgf #f7 ----- ---- #j #j ,9 ! fir/ ( ^!1 ! numb]s are writt5 at l5g?1 z we f\nd !m2 9 ! second1 z 9 7#gi71 unnecess>y ciph]s are /ruck (f1 & ! p]iods #fa1 #gf1 are n br\ght d[n1 until1 by ! cont9uance ( ! process1 !y cease to h ciph]s "u !m4 ,! foll[+ is ano!r example1 to :i* ! reason+ ( ! la/ article may be appli$4 #cd1hf1gh1dd1ja7ejjjj #cd1hf1gh1dd1ja7eijdi #be #jj #jj #jj #jj #ijjj #be -------------- #dj ---- #ajjjjj7 #i #hf #gh #dd #ja #i #aji7 #ihf #ijjj7 #i #ha #jj #jj #jj #iha ------------- ------- #ajjjjj7 #e #gh #dd #ja #aahjd7 #eghdd #ahjjj7 #d #gb #af #jj #dgbaf #dj7 ----------- -------- #ajjjjj7 #a #jf #bh #ja #aahjhi7ajfbhja #ahjjj7 #a #jf #bh #ja #ajfbhja #hj7 ---------- ------- #i7 #j #j #afc4 ,! rule is z foll[s3 ,to extract ! squ>e root ( a numb]2-- ,i4 ,beg9n+ f ! "r h&1 cut (f p]iods ( two figures ea*1 until n m ?an two are left4 ,,ii4 ,f9d ! root ( ! ne>e/ squ>e numb] next bel[ ! numb] 9 ! fir/ p]iod4 ,?is root is ! fir/ figure ( ! requir$ root2 subtract its squ>e f ! fir/ p]iod1 :i* gives ! fir/ rema9d]4 ,,iii4 ,annex ! second p]iod to ! "r ( ! rema9d]1 :i* gives ! fir/ divid5d4 ,,iv4 ,d\ble ! fir/ figure ( ! root2 see h[ (t5 ?is is conta9$ 9 ! numb] made by cutt+ "o figure f ! "r ( ! fir/ divid5d1 att5d+ to ,,ix41 if necess>y2 use ! quoti5t z ! second figure ( ! root2 annex x to ! "r ( ! d\ble ( ! fir/ figure1 & call ?is ! fir/ divisor4 ,v4 ,multiply ! fir/ divisor by ! second figure ( ! root2 if ! product be great] ?an ! fir/ divid5d1 use a l[] numb] = ! second figure ( ! root1 & = ! la/ figure ( ! divisor1 until ! multiplication j m5tion$ gives ! product less ?an ! fir/ divid5d2 subtract ?is f ! fir/ divid5d1 :i* gives ! second rema9d]4 ,,vi4 ,annex ! ?ird p]iod to ! second rema9d]1 :i* gives ! second divid5d4 ,,vii4 ,d\ble ! fir/ two figures ( ! root2#be see h[ (t5 ! result is conta9$ 9 ! numb] made by cutt+ "o figure f ! "r ( ! second divid5d2 use ! quoti5t z ! ?ird figure ( ! root2 annex x to ! "r ( ! d\ble ( ! fir/ two figures1 & call ?is ! second divisor4 #be ,or1 m simply1 add ! second figure ( ! root to ! fir/ divisor4 ,,viii4 ,get a new rema9d]1 z 9 ,v41 & repeat ! process until all ! p]iods are exhau/$2 if !re be !n no rema9d]1 ! squ>e root is f\nd2 if !re be a rema9d]1 ! propos$ numb] has no squ>e root1 & ! numb] f\nd z its squ>e root is ! squ>e root ( ! propos$ numb] dim9i%$ by ! rema9d]4 ,,ix4 ,:5 x happ5s t ! d\ble ( ! figures ( ! root is n conta9$ at all 9 all ! divid5d except ! la/ figure1 or :51 be+ conta9$ once1 #a is f\nd to give m ?an ! divid5d1 put a ciph] 9 ! squ>e root & 9 ! divisor1 & br+ d[n ! next p]iod2 %\ld ! same ?+ /ill happ51 put ano!r ciph] 9 ! root & divisor1 & br+ d[n ano!r p]iod2 & s on4 ,,ex]cises4 ,numb]s propos$4 ,squ>e roots4 #gcdda #bga #biibijj #agcj #fdadbdgiba #hjjhi #ijcfhghijfbe #iejfbe #dbdbjgdgdhbggfegf #bjeifbigf #acdbbfeicajaebdja #aaehefbja #afd4 ,s9ce ! squ>e ( a fraction is obta9$ by squ>+ ! num]ator & ! d5om9ator1 ! squ>e root ( a fraction is f\nd by tak+ ! squ>e root ( bo?4 ,?us1 ! squ>e root ( _/ is 1 s9ce #e #e is #be1 & #h #h is #fd4 ,if ! num]ator or d5om9ator1 or bo?1 be n squ>e numb]s1 x does n !re=e foll[ t ! fraction has no squ>e root2 = x may happ5 t multiplication or division by ! same numb] may conv]t bo? ! num]ator & d5om9ator into squ>e numb]s 7#ajh74 ,?us1 _/1 :i* appe>s at fir/ to h no squ>e root1 has "o 9 reality1 s9ce x is ! same z _/1 ^: squ>e root is 4 #afe4 ,we n[ proce$ f 7#aeh71 :]e x 0 /at$ t any numb] or fraction be+ giv51 a second may be f\nd1 ^: squ>e is z ne> to ! fir/ z we please4 ,?us1 ?\gh we _c solve ! problem1 8,f9d a fraction ^: squ>e is #b10 we c solve ! foll[+1 8,f9d a fraction ^: squ>e %all n diff] f #b by s mu* z #jjjjjjja40 ,in/ead ( ?is la/1 a /ill small] fraction may be sub/itut$2 9 fact1 any "o h[ev] small3 & 9 ?is process we are said to approximate to ! squ>e root ( #b4 ,?is c be done to any ext5t1 z foll[s3 ,suppose we wi% to f9d ! squ>e root ( #b )9 _/ ( ! tru?2 by :i* ,i mean1 to f9d a fraction a_/b ^: squ>e is less ?an #b1 b su* t ! squ>e ( a_/b _/ is great] ?an #b4 ,multiply ! num]ator & d5om9ator ( _/ by ! squ>e ( #eg1 or #cbdi1 :i* gives _/4 ,on attempt+ to extract ! squ>e root ( ! num]ator1 ,i f9d 7#afc7 t !re is a rema9d] #ih1 & t ! squ>e numb] next bel[ #fdih is #fdjj1 ^: root is #hj4 ,h5ce1 ! squ>e ( #hj is less ?an #fdih1 :ile t ( #ha is great]4 ,! squ>e root ( ! d5om9ator is ( c\rse #eg4 ,h5ce1 ! squ>e ( _/ is less ?an _/1 or #b1 :ile t ( _/ is great]1 & ^! two fractions only diff] by _/2 :i* 0 requir$ to be done4 #aff4 ,9 practice1 x is usual to f9d ! squ>e root true to a c]ta9 numb] ( places ( decimals4 ,?us1 #adadb is ! squ>e root ( #b true to f\r places ( decimals1 s9ce ! squ>e ( #adadb1 or #aiiiifafd1 is less ?an #b1 :ile an increase ( only #a 9 ! f\r? decimal place1 giv+ #adadc1 gives ! squ>e #bjjjbdddi1 :i* is great] ?an #b4 ,to take a m g5]al case3 ,suppose x requir$ to f9d ! squ>e root ( #afcg true to f\r places ( decimals4 ,! fraction is _/1 ^: squ>e root is to be f\nd )9 #jjja1 or _/4 ,annex ciph]s to ! num]ator & d5om9ator1 until ! d5om9ator becomes ! squ>e ( _/1 :i* gives _/1 extract ! squ>e root ( ! num]ator1 z 9 7#afc71 :i* %ews t ! squ>e numb] ne>e/ to x is #afcgjjjjj-acefd1 ^: root is #abgid4 ,h5ce1 _/1 or #abgid1 gives a squ>e less ?an #afcg1 :ile #abgie gives a squ>e great]4 ,9 fact1 ^! two squ>es are #afcfhfdcf & #afcgabjbe4 #afg4 ,! rule1 !n1 = extract+ ! squ>e root ( a numb] or decimal to any numb] ( places is3 ,annex ciph]s until !re are twice z _m places foll[+ ! units' place z !re are to be decimal places 9 ! root2 extract ! ne>e/ squ>e root ( ?is numb]1 & m>k (f ! giv5 numb] ( decimals4 ,or1 m simply3 ,divide ! numb] into p]iods1 s t ! units' figure %all be ! la/ ( a p]iod2 proce$ 9 ! usual way2 & if1 :5 decimals foll[ ! units' place1 !re is "o figure on ! "r1 9 a p]iod by itself1 annex a ciph] 9 br++ d[n t p]iod1 & aft]w>ds let ea* new p]iod consi/ ( two ciph]s4 ,place ! decimal po9t aft] t figure 9 =m+ :i* ! p]iod conta9+ ! units 0 us$4 #afh4 ,= example1 :at is ! squ>e root ( 7#a7 to five places ( decimals8 ,?is is 7#ade7 #acge1 & ! process is ! fir/ example ov] leaf4 ,! second example is ! extraction ( ! root ( #jha to sev5 places1 ! fir/ p]iod be+ #jh1 f :i* ! ciph] is omitt$ z useless4 #a1cg1e7aagbfj #a --- #ba7 #cg #ba ---- #bbg7afej #aehi ------ #bcdb7 #fajj #dfhd ------ #bcddf7adafjj #adjfgf -------- #bcdeb7 #ibdjj #h1a7bhdfjdi #d --- #dh7daj #chd ------ #efd7 #bfjj #bbef ------ #efhf7 #cddjj #cdaaf --------- #efibjd7 #bhdjjjj #bbgfhaf --------- #efibjh7 #efcahdjj #jjjjjbdacfgbbba7jjaeeceii #a --- #be7 #ada #abe ----- #cje7 #afcf #aebe ------ #cajc7 #aaagb #icji ------ #cajfe7 #ahfcbb #aeecbe -------- #cajgji7 #cjiigaj #bgifcha --------- #cjccbijj #afi4 ,:5 m ?an half ! decimals requir$ h be5 f\nd1 ! o!rs may be simply f\nd by divid+ ! divid5d by ! divisor1 z 9 7#aee74 ,! extraction ( ! squ>e root ( #ab to t5 places1 :i* w be f\nd 9 ! next page1 is an example4 ,x mu/1 h[ev]1 be obs]v$ 9 ?is process1 z 9 all o!rs :]e decimals are obta9$ by approximation1 t ! la/ place _c always be dep5d$ ^u3 on :i* acc\nt x is advisable to c>ry ! process s f>1 t "o or ev5 two m decimals %all be obta9$ ?an are absolutely requir$ to be correct4 ,a #ab7cdfdajafaeac #i --- #fd7 #cjj #bef ----- #fhf7 #ddjj #daaf ------ #fibd7 #bhdjj #bgfif ------ #fibha7 #gjdjj #fibha ---------- #fibhbja7 #aaaijjjj #fibhbja --------- #fibhbjbf7 #dbfagiijj #daefibaef ----------- #fibhbjcba7 #ajdhggddjj #fibhbjcba ----------- #fibhbjcbbe7 #ceeiedjgijj #cdfdajafabe ------------- #fibhbjcbcja7 #iedciaggejj #fibhbjcbcja ------------- #fibhbjcbcjbc7 #bfaegadeaiijj #bjghdfjifijfi ------------- #ecgbeceejhca ,b #fibhbjcbcjbf7 #ecgbeceejhca7ggedehgjedi #dhdigdbbfaah ------------ #ebbgicbdgac #dhdigdbbfaa ----------- #cghaijbajb #cdfdajafae ---------- #caghjjdhg #bggabhabi --------- #djfgbceh #cdfdajaf -------- #fjcacdb #eedbefb ------- #dhhghj #dhdigd ------ #chjf #cdfd ---- #cdb #bgg --- #fe #fb -- #c ,if f any rema9d] we cut (f ! ciph]s1 & all figures :i* w\ld come "u or on ! "r ( ^! ciph]s1 by a v]tical l9e1 we f9d on ! left ( t l9e a contract$ division1 su* z ^? 9 7#aee74 ,?us1 aft] hav+ f\nd ! root z f> z #cdfdaja1 we h ! rema9d] #dbfagii1 & ! divisor #fibhbjb4 ,! figures on ! left ( ! l9e are no?+ m ?an ! contract$ division ( ?is rema9d] by ! divisor1 ) ?is diff]5ce1 h[ev]1 t we h to beg9 by /rik+ a figure (f ! divisor1 in/ead ( us+ ! :ole divisor once1 & !n /rik+ (f ! fir/ figure4 ,by ?is alone we might h d\bl$ our numb] ( decimal places1 & got ! additional figures #faeacg1 ! la/ #g be+ obta9$ by c>ry+ ! contract$ division "o /ep fur!r ) ! rema9d] #ec4 ,we h1 !n1 ?is rule3 ,:5 half ! numb] ( decimal places h be5 obta9$1 in/ead ( annex+ two ciph]s to ! rema9d]1 /rike (f a figure f :at w\ld be ! divisor if ! process 7 cont9u$ at l5g?1 & divide ! rema9d] by ?is contract$ divisor1 z 9 7#aee74 ,z an example1 let u d\ble ! numb] ( decimal places already obta9$1 :i* are conta9$ 9 #cdfdajafaeac4 ,! rema9d] is #ecgbeceejhca1 ! divisor #fibhbjcbcjbf1 & ! process is z 9 7,b74 ,h5ce ! squ>e root ( #ab is1 #cdfdajafaeacggedehgjedi2 :i* is true to ! la/ figure1 & a little too great2 b ! sub/itution ( #h in/ead ( #i on ! "r h& w\ld make x too small4 ,,ex]cises4 ,numb]s4 ,squ>e roots4 #jjagbh #jdaefibaid #fdcd #hjbabbahe #hjgd #hiheedcid #aj #cafbbggff #aeg #abebiifdjhfadaffgghhdie ,,section ,,viii4 ,,on ,,! ,,proportion ,,( ,,numb]s4 #agj4 ,:5 two numb]s are nam$ 9 any problem1 x is usually necess>y1 9 "s way or o!r1 to comp>e ! two2 t is1 by consid]+ ! two toge!r1 to e/abli% "s connexion betwe5 !m1 :i* may be useful 9 future op]ations4 ,! fir/ me?od :i* sugge/s itself1 & ! mo/ simple1 is to obs]ve :i* is ! great]1 & by h[ mu* x diff]s f ! o!r4 ,! connexion ?us e/abli%$ betwe5 two numb]s may also hold good ( two o!r numb]s2 = example1 #h diff]s f #ai by #aa1 & #ajj diff]s f #aaa by ! same numb]4 ,9 ?is po9t ( view1 #h /&s to #ai 9 ! same situation 9 :i* #ajj /&s to #aaa1 ! fir/ ( bo? c\ples diff]+ 9 ! same degree f ! second4 ,! f\r numb]s ?us notic$1 viz43 #h1 #ai1 #ajj1 #aaa1 are said to be 9 ari?metical#bf proportion4 ,:5 f\r numb]s are ?us plac$1 ! fir/ & la/ are call$ ! extremes1 & ! second & ?ird ! means4 ,x is obvi\s t #aaa #h #ajj #ai1 t is1 ! sum ( ! extremes is equal to ! sum ( ! means4 ,& ?is is n accid5tal1 aris+ f ! p>ticul> numb]s we h tak51 b mu/ be ! case 9 e ari?metical proportion2 = 9 #aaa #h1 by 7#ce71 any dim9ution ( #aaa w n affect ! sum1 provid$ a correspond+ increase be giv5 to #h2 &1 by ! def9ition j giv51 "o mean is z mu* less ?an #aaa z ! o!r is great] ?an #h4 #bf ,?is is a v incorrect "n1 s9ce ! t]m #8ari?metical' applies equally to e notion 9 ?is book4 ,x is necess>y1 h[ev]1 t ! pupil %\ld use words 9 ! s5se 9 :i* !y w be us$ 9 8 succe$+ /udies4 #aga4 ,a set or s]ies ( numb]s is said to be 9 cont9u$ ari?metical proportion1 or 9 ari?metical progression1 :5 ! diff]5ce betwe5 e two succe$+ t]ms ( ! s]ies is ! same4 ,?is is ! case 9 ! foll[+ s]ies3 #a1 #b1 #c1 #d1 #e1 c4 #c1 #f1 #i1 #ab1 #ae1 c4 7#a71 #b1 7#b71 #c1 7#c71 c4 ,! diff]5ce betwe5 two succe$+ t]ms is call$ ! common diff]5ce4 ,9 ! ?ree s]ies j giv51 ! common diff]5ces are1 #a1 #c1 & 4 #agb4 ,if a c]ta9 numb] ( t]ms ( any ari?metical s]ies be tak51 ! sum ( ! fir/ & la/ t]ms is ! same z t ( any o!r two t]ms1 provid$ "o is z di/ant f ! beg9n+ ( ! s]ies z ! o!r is f ! end4 ,= example1 let !re be #g t]ms1 & let !m be1 a b c d e f g4 ,!n1 s9ce1 by ! nature ( ! s]ies1 b is z mu* above a z f is bel[ g 7#agj71 a g b f4 ,aga91 s9ce c is z mu* above b z e is bel[ f 7#agj71 b f c e4 ,b a g b f2 !re=e a g c e1 & s on4 ,aga91 twice ! middle t]m1 or ! t]m equally di/ant f ! beg9n+ & ! end 7:i* exi/s only :5 ! numb] ( t]ms is odd71 is equal to ! sum ( ! fir/ & la/ t]ms2 = s9ce c is z mu* bel[ d z e is above x1 we h c e d d #b;d4 ,b c e a g2 !re=e1 a g #b;d4 ,?is w give a %ort rule = f9d+ ! sum ( any numb] ( t]ms ( an ari?metical s]ies4 ,let !re be #g1 viz4 ^? j giv54 ,s9ce a g1 b f1 & c e1 are ! same1 _! sum is ?ree times 7a g71 :i* ) d1 ! middle t]m1 or half a g1 is ?ree times & a half 7a g71 or ! sum ( ! fir/ & la/ t]ms multipli$ by 7#c71 or _/1 or half ! numb] ( t]ms4 ,if !re _h be5 an ev5 numb] ( t]ms1 = example1 six1 viz4 a1 b1 c1 d1 e1 & f1 we "k n[ t a f1 b e1 & c d1 are ! same1 :5ce ! sum is ?ree times 7a f71 or ! sum ( ! fir/ & la/ t]ms multipli$ by half ! numb] ( t]ms1 z be=e4 ,! rule1 !n1 is3 ,to sum any numb] ( t]ms ( an ari?metical progression1 multiply ! sum ( ! fir/ & la/ t]ms by half ! numb] ( t]ms4 ,= example1 :at are #ii t]ms ( ! s]ies #a1 #b1 #c1 c48 ,! #ii? t]m is #ii1 & ! sum is #ii #ajj #ii 7#ii #a7---1 or --------1 or #diej4 #b #b ,! sum ( #ej t]ms ( ! s]ies #a #b #d #e 7 #a #ej 7 #ej ---1 ---1 #a1 ---1 ---1 #b1 c4 is 7--- ---7---1 #c #c #c #c 7 #c #c 7 #b or #ag #be1 or #dbe4 #agc4 ,! fir/ t]m be+ giv51 & also ! common diff]5ce & numb] ( t]ms1 ! la/ t]m may be f\nd by add+ to ! fir/ t]m ! common diff]5ce multipli$ by "o less ?an ! numb] ( t]ms4 ,= x is evid5t t ! second t]m diff]s f ! fir/ by ! common diff]5ce1 ! ?ird t]m by twice1 ! f\r? t]m by ?ree times ! common diff]5ce2 & s on4 ,or1 ! passage f ! fir/ to ! n? t]m is made by n-#a /eps1 at ea* ( :i* ! common diff]5ce is add$4 ,,ex]cises4 ,giv54 ,to f9d4 ,s]ies4 ,no4 ( t]ms4 ,la/ t]m4 ,sum4 #d1 7#f71 #i1 c4 #cc #hd #adeb #a1 #c1 #e1 c4 #bh #ee #ghd #b1 #bj1 #ch1 c4 #ajj1jjj #agiiihd #hiiiicjjjjj #agd4 ,! sum be+ giv51 ! numb] ( t]ms1 & ! fir/ t]m1 we c !nce f9d ! common diff]5ce4 ,suppose1 = example1 ! fir/ t]m ( a s]ies to be "o1 ! numb] ( t]ms #ajj1 & ! sum #aj1jjj4 ,s9ce #aj1jjj 0 made by multiply+ ! sum ( ! fir/ & la/ t]ms by _/1 if we divide by ?is1 we %all recov] ! sum ( ! fir/ & la/ t]ms4 ,n[1 1_/ divid$ by _/ is 7#abb7 #bjj1 & ! fir/ t]m be+ #a1 ! la/ t]m is #aii4 ,we h !n to pass f #a to #aii1 or ?r\gh #aih1 by #ii equal /eps4 ,ea* /ep is1 !re=e1 _/1 or #b1 :i* is ! common diff]5ce2 or ! s]ies is #a1 #c1 #e1 c41 up to #aii4 ,giv54 ,to f9d4 ,sum4 ,no4 ( t]ms4,fir/ t]m4,la/ t]m4,common diff4 #ahjijbe #acde #a #bfhi #b #dd #aj #c _/ _/ #gjgefjj #accj #d #ajfcf #h #age4 ,we n[ return to 7#agj71 9 :i* we comp>$ two numb]s toge!r by _! diff]5ce4 ,?is1 h[ev]1 is n ! me?od ( comp>ison :i* we employ 9 common life1 z any s+le famili> in/ance w %ew4 ,= example1 we say ( ,a1 :o has #aj ?\s& p\nds1 t he is mu* ri*] ?an ,b1 :o has only #c ?\s&2 b we d n say t ,c1 :o has #ajg ?\s& p\nds1 is mu* ri*] ?an ,d1 :o has #ajj ?\s&1 ?\gh ! diff]5ce ( =tune is ! same 9 bo? cases1 viz4 #g ?\s& p\nds4 ,9 comp>+ numb]s we take into our reckon+ n only ! diff]5ces1 b ! numb]s !mselves4 ,?us1 if ,b & ,d bo? receiv$ #g ?\s& p\nds1 ,b w\ld receive #bcc p\nds & a ?ird = e #ajj p\nds :i* he _h be=e1 :ile ,d = e #ajj p\nds w\ld receive only #g p\nds4 ,& ?\gh1 9 ! view tak5 9 7#agj71 #c is z ne> to #aj z #ajj is to #ajg1 yet1 9 ! light 9 :i* we n[ reg>d !m1 #c is n s ne> to #aj z #ajj is to #ajg1 = #c diff]s f #aj by m ?an twice itself1 :ile #ajj does n diff] f #ajg by s mu* z "o-fif? ( itself4 ,?is is express$ 9 ma!matical language by say+1 t ! ratio or proportion ( #aj to #c is great] ?an ! ratio or proportion ( #ajg to #ajj4 ,we proce$ to def9e ^! t]ms m accurately4 #agf4 ,:5 we use ! t]m "p ( a numb] or fraction 9 ! rema9d] ( ?is section1 we mean1 "o ( ! v>i\s sets ( equal p>ts into :i* x may be divid$1 ei!r ! half1 ! ?ird1 ! f\r?1 c43 ! t]m multiple has be5 already expla9$ 7#ajb74 ,by ! t]m multiple-"p ( a numb] we mean1 ! abbreviation ( ! words multiple ( a "p4 ,?us1 #a1 #b1 #c1 #d1 & #f1 are p>ts ( #ab2 is also a "p ( #ab1 be+ conta9$ 9 x #bd times2 #ab1 #bd1 #cf1 c41 are multiples ( #ab2 & #h1 #i1 _/1 c4 are multiple p>ts ( #ab1 be+ multiples ( "s ( its p>ts4 ,& :5 multiple p>ts g5]ally are spok5 (1 ! p>ts !mselves are suppos$ to be includ$1 on ! same pr9ciple t #ab is c\nt$ among ! multiples ( #ab1 ! multipli] be+ #a4 ,! multiples !mselves are also includ$ 9 ?is t]m2 = #bd is also #dh halves1 & is !re=e among ! multiple p>ts ( #ab4 ,ea* "p is also 9 v>i\s ways a multiple-"p2 = "o-f\r? is two-eigh?s1 & ?ree-twelf?s1 c4 #agg4 ,e numb] or fraction is a multiple-"p ( e o!r numb] or fraction4 ,if1 = example1 we ask :at "p #ab is ( #g1 we see t on divid+ #g into #g p>ts1 & repeat+ "o ( ^! p>ts #ab times1 we obta9 #ab2 or1 on divid+ #g into #ad p>ts1 ea* ( :i* is "o-half1 & repeat+ "o ( ^! p>ts #bd times1 we obta9 #bd halves1 or #ab4 ,h5ce1 #ab is _/1 or _/1 or _/ ( #g2 & s on4 ,g5]ally1 :5 a & b are two :ole numb]s1 a_/b expresses ! multiple-"p :i* a is ( b1 & b_/a t :i* b is ( a4 ,aga91 suppose x requir$ to det]m9e :at multiple-"p 7#b7 is ( 7#c71 or _/ ( _/4 ,^! fractions1 r$uc$ to a common d5om9ator1 are _/ & _/1 ( :i* ! second1 divid$ into #aab p>ts1 gives _/1 :i* repeat$ #ge times gives _/1 ! fir/4 ,h5ce1 ! multiple-"p :i* ! fir/ is ( ! second is _/1 :i* be+ obta9$ by ! rule giv5 9 7#aba71 %ews t a_/b1 or a divid$ by b1 accord+ to ! notion ( division !re giv51 expresses ! multiple-"p :i* a is ( b 9 e case4 #agh4 ,:5 ! fir/ ( f\r numb]s is ! same multiple-"p ( ! second :i* ! ?ird is ( ! f\r?1 ! f\r are said to be geometrically#bg proportional1 or simply proportional4 ,?is is a ^w 9 common use2 & x rema9s to %ew t our ma!matical def9ition ( x1 j giv51 is1 9 fact1 ! common notion atta*$ to x4 ,= example1 suppose a picture is copi$ on a small] scale1 s t a l9e ( two in*es long 9 ! orig9al is repres5t$ by a l9e ( "o in* & a half 9 ! copy2 we say t ! copy is n correct unless all ! p>ts ( ! orig9al are r$uc$ 9 ! same proportion1 namely1 t ( #b to 7#a74 ,s9ce1 on divid+ two in*es into #d p>ts1 & tak+ #c ( !m1 we get 7#a71 ! same mu/ be done ) all ! l9es 9 ! orig9al1 t is1 ! l5g? ( any l9e 9 ! copy mu/ be ?ree p>ts out ( f\r ( its l5g? 9 ! orig9al4 ,aga91 int]e/ be+ at #e p] c5t1 t is1 #e be+ giv5 = ! use ( #ajj1 a simil> proportion ( e o!r sum w\ld be giv52 ! int]e/ ( #gj1 = example1 w\ld be j su* a "p ( #gj z #e is ( #ajj4 #bg ,! same rem>k may be made "h z 0 made 9 ! note on ! t]m #8ari?metical proportion1' page #aja4 ,! ^w #8geometrical' is1 g5]ally speak+1 dropp$1 except :5 we wi% to di/+ui% betwe5 ?is k9d ( proportion & t :i* has be5 call$ ari?metical4 ,s9ce1 !n1 ! "p :i* a is ( b is express$ by ! fraction a_/b1 or any o!r fraction :i* is equival5t to x1 & t :i* c is ( d by c_/d1 x foll[s1 t :5 a1 b1 c1 & d1 are proportional1 a_/b c_/d4 ,?is equation w be ! f\ndation ( all our reason+ on proportional quantities2 & 9 consid]+ proportionals1 x is necess>y to obs]ve n only ! quantities !mselves1 b also ! ord] 9 :i* !y come4 ,?us1 a1 b1 c1 & d1 be+ proportionals1 t is1 a be+ ! same multiple-"p ( b :i* c is ( d1 x does n foll[ t a1 d1 b1 & c are proportionals1 t is1 t a is ! same multiple-"p ( d :i* b is ( c4 ,x is pla9 t a is great] ?an1 equal to1 or less ?an b1 accord+ z c is great] ?an1 equal to1 or less ?an d4 #agi4 ,f\r numb]s1 a1 b1 c1 & d1 be+ proportional 9 ! ord] writt51 a & d are call$ ! extremes1 & b & c ! means1 ( ! proportion4 ,= conv5i5ce1 we w call ! two extremes1 or ! two means1 simil> t]ms1 & an extreme & a mean1 dissimil> t]ms4 ,?us1 a & d are simil>1 & s are b & c2 :ile a & b1 a & c1 d & b1 d & c1 are dissimil>4 ,x is cu/om>y to express ! proportion by plac+ dots betwe5 ! numb]s1 ?us3 a 3 b c 3 d #ahj4 ,equal numb]s w /ill rema9 equal :5 !y h be5 increas$1 dim9i%$1 multipli$1 or divid$1 by equal quantities4 ,?is am\nts to say+ t if a b & p q1 a p b q1 a - p b - q1 ap bq1 a b & --- ---4 p q ,x is also evid5t1 t a p-p1 a -p p1 ap_/p1 & a_/p p1 are all equal to a4 #aha4 ,! product ( ! extremes is equal to ! product ( ! means4 ,let a_/b c_/d1 & multiply ^! equal numb]s by ! product bd4 ,!n1 a abd --- bd ----- 7#aaf7 ad1 b b c cbd & --- bd ----- cb3 h5ce 7#ahj71 ad bc4 d d ,?us1 #f1 #h1 #ba1 & #bh1 are proportional1 s9ce #f #c #c #g #ba --- --- ------ --- 7#ahj72 #h #d #d #g #bh & x appe>s t #f #bh #h #ba1 s9ce bo? products are #afh4 #ahb4 ,if ! product ( two numb]s be equal to ! product ( two o!rs1 ^! numb]s are proportional 9 any ord] :atev]1 provid$ ! numb]s 9 ! same product are s plac$ z to be simil> t]ms2 t is1 if ab pq1 we h ! foll[+ proportions3-- a 3 p q 3 b a 3 q p 3 b b 3 p q 3 a b 3 q p 3 a p 3 a b 3 q p 3 b a 3 q q 3 a b 3 p q 3 b a 3 p ,to prove any "o ( ^!1 divide bo? ab & pq by ! product ( its second & f\r? t]ms2 = example1 to %ew ! tru? ( a3 q p3 b1 divide bo? ab & pq by bq4 ,!n1 ab a pq p ---- ---1 & ---- ---2 h5ce 7#ahj71 bq q bq b a p --- ---1 or a 3 q p 3 b4 q b ,! pupil %\ld n fail to prove e "o ( ! eight cases1 & to v]ify !m by "s simple examples1 su* z #a #f #b #c1 :i* gives #a3 #b #c3 #f1 #c3 #a #f3 #b1 c4 #ahc4 ,h5ce1 if f\r numb]s be proportional1 !y are also proportional 9 any o!r ord]1 provid$ x be su* t simil> t]ms /ill rema9 simil>4 ,= s9ce1 :5 a c --- ---1 b d x foll[s 7#aha7 t ad bc1 all ! proportions :i* foll[ f ad bc1 by ! la/ article1 foll[ also f a c --- ---1 b d #ahd4 ,f 7#aad7 x foll[s t a b a #a --- ---------1 b b a & if --- be less ?an #a1 b a b - a #a - --- ---------1 b b a :ile if --- be great] ?an #a1 b a a - b --- - #a ---------4 b b a b a - b ,also 7#abb71 if --------- be divid$ by --------- b b a b ! result is ---------4 a - b ,h5ce1 a1 b1 c1 & d1 be+ proportionals1 we may obta9 o!r proportions1 ?us3 a c ,let --- --- b d a c ,!n 7#aad7 #a --- #a --- b d a b c d or --------- --------- b d or a b3 b c d3 d ,t is1 ! sum ( ! fir/ & second is to ! second z ! sum ( ! ?ird & f\r? is to ! f\r?4 ,= brevity1 we %all n /ate 9 words any m ( ^! proportions1 s9ce ! pupil w easily supply :at is want+4 ,resum+ ! proportion a3 b c3 d a c or --- --- b d a c a #a - --- #a - ---1 if --- be less ?an #a1 b d b b - a d - c or --------- --------- b d t is1 b-a3 b d-c3 d or1 a-b3 b c-d3 d1 a if --- be great] ?an #a4 b a b c d ,aga91 s9ce --------- --------- b d a - b c - d a & --------- --------- 7--- be+ great] ?an #a7 b d b a b c d divid+ ! fir/ by ! second we h --------- ----------1 a - b c - d or a b 3 a - b c d 3 c - d & also a b 3 b - a c d 3 d - c1 a if --- be less ?an #a4 b #ahe4 ,_m o!r proportions might be obta9$ 9 ! same mann]4 ,we w1 h[ev]1 cont5t ourselves ) writ+ d[n a few :i* c be obta9$ by comb9+ ! prec$+ articles4 a b 3 a c d 3 c a 3 a - b c 3 c - d a c 3 a - c b d 3 b - d4 ,9 ^! & all o!rs x mu/ be obs]v$1 t :5 su* expressions z a-b & c-d occur1 x is suppos$ t a is great] ?an b1 & c great] ?an d4 #ahf4 ,if f\r numb]s be proportional1 & any two dissimil> t]ms be bo? multipli$1 or bo? divid$ by ! same quantity1 ! results are proportional4 ,?us1 if a3 b c3 d1 & m & n be any two numb]s1 we h also ! foll[+3 ma 3 b mc 3 d a 3 mb c 3 md a c --- 3 mb --- 3 md n n ma 3 nb mc 3 nd a b c d --- 3 --- --- 3 --- m m m m a b c d --- 3 --- --- 3 --- m m n n & v>i\s o!rs4 ,to prove any "o ( ^!1 recollect t no?+ m is necess>y to make f\r numb]s proportional except t ! product ( ! extremes %\ld be equal to t ( ! means4 ,take ! ?ird ( ^? j giv52 ! product ( its extremes is a mad --- md1 or -----1 n n c mbc :ile t ( ! means is mb ---1 or -----4 n n ,b s9ce a 3 b c 3 d1 by 7#aha7 ad bc1 mad mbc :5ce1 by 7#ahj71 mad mbc1 & ----- -----4 n n a c ,h5ce1 ---1 mb1 ---1 & md1 are proportionals4 n n #ahg4 ,if ! t]ms ( "o proportion be multipli$ by ! t]ms ( a second1 ! products are proportional2 t is1 if a3 b c3 d1 & p3 q r3 s1 x foll[s t ap3 bq cr3 ds4 ,=1 s9ce ad bc1 & ps qr1 by 7#ahj7 adps bcqr1 or ap ds bq cr1 :5ce 7#ahb7 ap3 bq cr3 ds4 #ahh4 ,if f\r numb]s be proportional1 any simil> p[]s ( ^! numb]s are also proportional2 t is1 if a 3 b c 3 d ,!n aa 3 bb cc 3 dd aaa 3 bbb ccc 3 ddd c4 c4 ,=1 if we write ! proportion twice1 ?us1 a 3 b c 3 d a 3 b c 3 d by 7#ahg7 aa 3 bb cc 3 dd ,b a 3 b c 3 d ,:5ce 7#ahg7 aaa 3 bbb ccc 3 ddd2 & s on4 #ahi4 ,an expression is said to be homog5e\s ) respect to any two or m lett]s1 = in/ance1 a1 b1 & c1 :5 e t]m ( x conta9s ! same numb] ( lett]s1 c\nt+ a1 b1 & c only4 ,?us1 maab nabc rccc is homog5e\s ) respect to a1 b1 & c2 & ( ! ?ird degree1 s9ce 9 ea* t]m !re is ei!r a1 b1 & c1 or "o ( ^! repeat$ alone1 or ) ano!r1 s z to make ?ree 9 all4 ,?us1 #h;aaabc1 #ab;abccc1 maaaaa1 naabbc1 are all homog5e\s1 & ( ! fif? degree1 ) respect to a1 b1 & c only2 & any expression made by add+ or subtract+ ^! f "o ano!r1 w be homog5e\s & ( ! fif? degree4 ,aga9 ma mnb is homog5e\s ) respect to a & b1 & ( ! fir/ degree2 b x is n homog5e\s ) respect to m & n1 ?\gh x is s ) respect to a & n4 ,?is be+ premis$1 we proce$ to a !orem1#bh :i* w conta9 all ! results ( 7#ahd71 7#ahe71 & 7#ahh74 #bh ,a !orem is a g5]al ma!matical fact3 ?us1 t e numb] is divisible by f\r :5 its la/ two figures are divisible by f\r1 is a !orem2 t 9 e proportion ! product ( ! extremes is equal to ! product ( ! means1 is ano!r4 #aij4 ,if any f\r numb]s be proportional1 & if f ! fir/ two1 a & b1 any two homog5e\s expressions ( ! same degree be =m$2 & if f ! la/ two1 two o!r expressions be =m$1 9 precisely ! same mann]1 ! f\r results w be proportional4 ,= example1 if a3 b c3 d1 & if #b;aaa #c;aab & bbb abb be *os51 :i* are bo? homog5e\s ) respect to a & b1 & bo? ( ! ?ird degree2 & if ! correspond+ expressions #b;ccc #c;ccd & ddd cdd be =m$1 :i* are made f c & d precisely 9 ! same mann] z ! two =m] ones f a & b1 !n w #b;aaa #c;aab 3 bbb abb #b;ccc #c;ccd 3 ddd cdd a ,to prove ?is1 let --- be call$ x4 b a a c ,!n1 s9ce --- x1 & --- ---1 b b d c x foll[s t --- x4 d ,b s9ce a divid$ by b gives x1 x multipli$ by b w give a1 or a bx4 ,= a simil> reason1 c dx4 ,put bx & dx in/ead ( a & c 9 ! f\r expressions j giv51 recollect+ t :5 quantities are multipli$ toge!r1 ! result is ! same 9 :atev] ord] ! multiplications are made2 t1 = example1 bxbxbx is ! same z bbbxxx4 ,h5ce1 #b;aaa #c;aab #b;bxbxbx #c;bxbxb #b;bbbxxx #c;bbbxx :i* is bbb multipli$ by #bxxx #cxx or bbb 7#bxxx #cxx7#bi ,simil>ly1 #b;ccc #c;ccd ddd 7#bxxx #cxx7 ,also1 bbb abb bbb bxbb bbb multipli$ by #a x or bbb7#a x7 ,simil>ly1 ddd cdd ddd 7#a x7 ,n[1 bbb 3 bbb ddd 3 ddd #bi ,if bx be sub/itut$ = a 9 any expression :i* is homog5e\s ) respect to a & b1 ! pupil may easily see t b mu/ occur 9 e t]m z (t5 z !re are units 9 ! degree ( ! expression3 ?us1 aa ab becomes bxbx bxb or bb7xx x72 aaa bbb becomes bxbxbx bbb or bbb7xxx #a72 & s on4 ,:5ce 7#ahf71 bbb7#bxxx #cxx73 bbb7#a x7 ddd7#bxxx #cxx73 ddd7#a x71 :i*1 :5 in/ead ( ^! expressions _! equals j f\nd are sub/itut$1 becomes #b;aaa #c;aab3 bbb abb #b;ccc #c;ccd3 ddd cdd4 ,! same reason+ may be appli$ to any o!r case1 & ! pupil may 9 ?is way prove ! foll[+ !orems3 ,if a 3 b c 3 d #b;a #c;b 3 b #b;c #c;d 3 d aa bb 3 aa - bb cc dd 3 cc - dd mab 3 #b;aa bb mcd 3 #b;cc dd #aia4 ,if ! two means ( a proportion be ! same1 t is1 if a 3 b b3 c1 ! ?ree numb]s1 a1 b1 & c1 are said to be 9 cont9u$ proportion1 or 9 geometrical progression4 ,! same t]ms are appli$ to a s]ies ( numb]s1 ( :i* any ?ree t foll[ "o ano!r are 9 cont9u$ proportion1 su* z #a #b #d #h #af #cb #fd c4 #b #b #b #b #b #b #b --- --- --- --- ---- ---- c4 #c #i #bg #ha #bdc #gbi ,:i* are 9 cont9u$ proportion1 s9ce #b #b #b #a 3 #b #b 3 #d #b 3 --- --- 3 --- #c #c #i #b #b #b #b #b 3 #d #d 3 #h --- 3 --- --- 3 --- #c #i #i #bg c4 c4 #aib4 ,let a1 b1 c1 d1 e be 9 cont9u$ proportion2 we h !n a b a 3 b b 3 c or --- --- or ac bb b c b c b 3 c c 3 d --- --- bd cc c d c d c 3 d d 3 e --- --- ce dd d e ,ea* t]m is =m$ f ! prec$+1 by multiply+ x by ! same numb]4 ,?us1 b c b --- a 7#ahj72 c --- b2 a b a b b c b & s9ce --- ---1 --- --- or c --- b4 b c a b a d d c b ,aga91 d --- c1 b --- ---1 :i* is ---2 c c b a b !re=e1 d --- c1 & s on4 c b ,if1 !n1 --- a 7:i* is call$ ! common ratio ( ! s]ies7 be d5ot$ by r1 we h b ar c br arr d cr arrr & s on2 :5ce ! s]ies a b c d c4 is a ar arr arrr c4 ,h5ce a 3 c a 3 arr 7#ahf7 aa 3 a>r aa 3 bb because1 b be+ ar1 bb is ar> or a>r4 ,aga91 a 3 d a 3 arrr 7#ahf7 aaa 3 aa>rr aaa 3 bbb ,also a 3 e aaaa 3 bbbb1 & s on2 t is1 ! fir/ be>s to ! n t]m f ! fir/ ! same proportion z ! n p[] ( ! fir/ to ! n p[] ( ! second4 #aic4 ,a %ort rule may be f\nd = add+ toge!r any numb] ( t]ms ( a cont9u$ proportion4 ,let x be fir/ requir$ to add toge!r ! t]ms #a1 r1 rr1 c4 :]e r is great] ?an unity4 ,x is evid5t t we d n alt] any expression by add+ or subtract+ any numb]s1 provid$ we aft]w>ds subtract or add ! same4 ,= example1 p p-q q-r r- s s ,let u take f\r t]ms ( ! s]ies1 #a1 r1 rr1 c4 or1 #a r rr rrr ,x is pla9 t rrrr-#a rrrr-rrr rrr-rr rr-r r-#a ,n[ 7#ed71 rr-r r7r-#a71 rrr -rr rr7r-#a71 rrrr-rrr rrr7r-#a71 & ! above equation becomes rrrr -#a rrr7r-#a7 rr 7r-#a7 r 7r-#a7 r-#a2 :i* is 7#ed7 rrr rr r #a tak5 r-#a times4 ,h5ce1 rrrr-#a divid$ by r-#a w give #a r rr rrr1 ! sum ( ! t]ms requir$4 ,9 ?is way may be prov$ ! foll[+ s]ies ( equations3 rr - #a #a r -------- r - #a rrr - #a #a r rr --------- r - #a rrrr - #a #a r rr rrr ---------- r - #a rrrrr - #a #a r rr rrr rrrr ----------- r - #a ,if r be less ?an unity1 9 ord] to f9d #a r rr rrr1 obs]ve t #a - rrrr #a - r r - rr rr - rrr rrr - rrrr #a - r r7#a - r7 rr7#a - r7 rrr7#a - r72 :5ce1 by simil> reason+1 #a r rr rrr is f\nd by divid+ #a-rrrr by #a-r2 & equations simil> to ^! j giv5 may be f\nd1 :i* are1 #a - rr #a r -------- #a - r #a - rrr #a r rr --------- #a - r #a - rrrr #a r rr rrr ---------- #a - r #a - rrrrr #a r rr rrr rrrr ----------- #a - r ,! rule is3 ,to f9d ! sum ( n t]ms ( ! s]ies1 #a r rr c41 divide ! diff]5ce betwe5 #a & ! 7n #a7 t]m by ! diff]5ce betwe5 #a & r4 #aid4 ,?is may be appli$ to f9d+ ! sum ( any numb] ( t]ms ( a cont9u$ proportion4 ,let a1 b1 c1 c4 be ! t]ms ( :i* x is requir$ to sum f\r1 t is1 to f9d a b c d1 or 7#aib7 a ar arr arrr1 or 7#ed7 a7#a r rr rrr71 :i* 7#aic7 is rrrr - #a #a - rrrr ---------- a1 or ---------- a1 r - #a #a - r accord+ z r is great] or less ?an unity4 ,! fir/ fraction is arrrr - a e - a -------------1 or 7#aib7 ---------4 r - #a r - #a a - e ,simil>ly1 ! second is ---------4 #a - r ,! rule1 !re=e1 is3 ,to sum n t]ms ( a cont9u$ proportion1 divide ! diff]5ce ( ! 7n #a7 & fir/ t]ms by ! diff]5ce betwe5 unity & ! common measure4 ,= example1 ! sum ( #aj t]ms ( ! s]ies #a #c #i #bg c4 is requir$4 ,! elev5? t]m is #eijdi1 & _/ is #biebd4 ,aga91 ! sum ( #ah t]ms ( ! s]ies #b #a c4 ( :i* ! n9ete5? t]m is _/1 is #a #b - ------ #acajgb #acajgj ----------- #c ------4 #a - #acajgb ,,examples4 #i t]ms ( #a #d #af c4 are #hgcha #f #ab #hdgdbbfge #aj 444444 #c --- ---- c4 444 --------- #g #di #bjagfhjce #a #a #a #ajdhege #bj 444444 --- --- --- c4 444 ------- #b #d #h #ajdhegf #aie4 ,! p[]s ( a numb] or fraction great] ?an unity increase2 = s9ce #b is great] ?an #a1 #b #b is #b tak5 m ?an once1 t is1 is great] ?an #b1 & s on4 ,?is increase goes on )\t limit2 t is1 !re is no quantity s great b t "s p[] ( #b is great]4 ,to prove ?is1 obs]ve t e p[] ( #b is made by multiply+ ! prec$+ p[] by #b1 or by #a #a1 t is1 by add+ to ! =m] p[] t p[] itself & its half4 ,!re w1 !re=e1 be m add$ to ! #aj? p[] to =m ! #aa?1 ?an 0 add$ to ! #i? p[] to =m ! #aj?4 ,b x is evid5t t if any giv5 quantity1 h[ev] small1 be cont9ually add$ to #b1 ! result w come 9 "t to exce$ any o!r quantity t 0 also giv51 h[ev] great2 mu* m1 !n1 w x d s if ! quantity add$ to #b be increas$ at ea* /ep1 :i* is ! case :5 ! successive p[]s ( #b are =m$4 ,x is evid5t1 also1 t ! p[]s ( #a nev] increase1 be+ always #a2 ?us1 #a #a #a1 c4 ,also1 if a be great] ?an m times b1 ! squ>e ( a is great] ?an mm times ! squ>e ( b4 ,?us1 if a #b;b c1 :]e a is great] ?an #b;b1 ! squ>e ( a1 or aa1 :i* is 7#fh7 #d;bb #d;bc cc is great] ?an #d;bb1 & s on4 #aif4 ,! p[]s ( a fraction less ?an unity cont9ually decrease2 ?us1 ! squ>e ( 1 or 1 is less ?an 1 be+ only two-fif?s ( x4 ,?is decrease cont9ues )\t limit2 t is1 !re is no quantity s small b t "s p[] ( is less4 ,= if #e #b #a #a #a --- x1 --- ---1 & ! p[]s ( are ----1 -----1 #b #e x xx xxx & s on4 ,s9ce x is great] ?an #a 7#aie71 "s p[] ( x may be f\nd :i* %all be great] ?an a giv5 quantity4 ,let ?is be call$ m2 !n #a_/m is ! correspond+ p[] ( 2 & a fraction ^: d5om9ator c be made z great z we please1 c itself be made z small z we please 7#aab74 #aig4 ,we h1 !n1 9 ! s]ies #a r rr rrr rrrr c4 ,i4 ,a s]ies ( increas+ t]ms1 if r be great] ?an #a4 ,,ii4 ,( t]ms hav+ ! same value1 if r be equal to #a4 ,,iii4 ,a s]ies ( decreas+ t]ms1 if r be less ?an #a4 ,9 ! fir/ two cases1 ! sum #a r rr rrr c4 may evid5tly be made z great z we please1 by suffici5tly increas+ ! numb] ( t]ms4 ,b 9 ! ?ird ?is may or may n be ! case2 = ?\gh some?+ is add$ at ea* /ep1 yet1 z t augm5tation dim9i%es at e /ep1 we may n c]ta9ly say t we c1 by any numb] ( su* augm5tations1 make ! result z great z we please4 ,to %ew ! contr>y 9 a simple in/ance1 consid] ! s]ies1 #a _/ c4 ,c>ry ?is s]ies to :at ext5t we may1 x w always be necess>y to add ! la/ t]m 9 ord] to make z mu* z #b4 ,?us1 7#a 7 #a #a #a #b 7#a 7 #b4 7#a _/7 _/ #b1 c4 ,b 9 ! s]ies1 e t]m is only ! half ( ! prec$+2 consequ5tly no numb] ( t]ms1 h[ev] great1 c be made z great z #b by add+ "o m4 ,! sum1 !re=e1 ( #a1 1 1 c4 cont9ually approa*es to #b1 dim9i%+ its di/ance f #b at e /ep1 b nev] rea*+ x4 ,h5ce1 #b is cell$ ! limit ( #a c4 ,we are n1 !re=e1 to conclude t e s]ies ( decreas+ t]ms has a limit4 ,! contr>y may be %ewn 9 ! v simple s]ies1 #a c4 :i* may be writt5 ?us3 #a 7 7 7 444 up to 7 7 444 up to _/7 7_/ 444 up to _/7 c4 ,we h ?us divid$ all ! s]ies1 except ! fir/ two t]ms1 into lots1 ea* conta9+ half z _m t]ms z !re are units 9 ! d5om9ator ( its la/ t]m4 ,?us1 ! f\r? lot conta9s #af or _/#b t]ms4 ,ea* ( ^! lots may be %ewn to be great] ?an 4 ,take ! ?ird1 = example1 consi/+ ( 1 _/1 _/1 _/1 _/1 _/1 _/1 & _/4 ,all except _/1 ! la/1 are great] ?an _/2 consequ5tly1 by sub/itut+ _/ = ea* ( !m1 ! am\nt ( ! :ole lot w\ld be less5$2 & z x w\ld !n become _/1 or 1 ! lot itself is great] ?an 4 ,n[1 if to #a 1 be cont9ually add$1 ! result w 9 "t exce$ any giv5 numb]4 ,/ill m w ?is be ! case if1 in/ead ( 1 ! sev]al lots writt5 above be add$ "o aft] ! o!r4 ,b x is ?us t ! s]ies #a 1 c4 is compos$1 :i* proves :at 0 said1 t ?is s]ies has no limit4 #aih4 ,! s]ies #a r rr rrr c4 always has a limit :5 r is less ?an #a4 ,to prove ?is1 let ! t]m succe$+ t at :i* we /op be a1 :5ce 7#aid7 ! sum is #a - a #a a -------1 or 7#aab7 ------- - ------4 #a - r #a - r #a - r ,! t]ms decrease )\t limit 7#aif71 :5ce we may take a t]m s f> di/ant f ! beg9n+1 t a1 & !re=e a -------1 #a - r %all be z small z we please4 ,b x is evid5t t 9 ?is case #a a ------- - ------- ?\gh always less ?an #a - r #a - r #a #a -------- may be br\ght z ne> to ------- #a - r #a - r z we please2 t is1 ! s]ies #a r rr c4 cont9ually approa*es to ! limit #a --------4 #a - r ,?us #a c4 :]e r 1 cont9ually approa*es to #a ----- or #b1 z 0 %ewn 9 ! la/ article4 #a - ,,ex]cises4 #b #b ,! limit ( #b --- --- c4 #c #i #a #a or #b7a --- --- c47 is #c #c #i #i #ha 444 #a --- ---- c4 444 #aj #aj #ajj #ae #de 444 #e ---- ---- c4 444 #h #g #di #aii4 ,:5 ! fraction a_/b is n equal to c_/d1 b great]1 a is said to h to b a great] ratio ?an c has to d2 & :5 a_/b is less ?an c_/d1 a is said to h to b a less ratio ?an c has to d4 ,we propose ! foll[+ que/ions z ex]cises1 s9ce !y foll[ v simply f ?is def9ition4 ,i4 ,if a be great] ?an b1 & c less ?an or equal to d1 a w h a great] ratio to b ?an c has to d4 ,,ii4 ,if a be less ?an b1 & c great] ?an or equal to d1 a has a less ratio to b ?an c has to d4 ,,iii4 ,if a be to b z c is to d1 & if a h a great] ratio to b ?an c has to x1 d is less ?an x2 & if a h a less ratio to b ?an c to x1 d is great] ?an x4 ,,iv4 a has to b a great] ratio ?an ax to bx y1 & a less ratio ?an ax to bx- y4 #bjj4 ,if a h to b a great] ratio ?an c has to d1 a c has to b d a less ratio ?an a has to b1 b a great] ratio ?an c has to d2 or1 9 o!r words1 if a_/b be ! great] ( ! two fractions a_/b & c_/d1 a c --------- b d w be great] ?an c_/d1 b less ?an a_/b4 ,to %ew ?is1 obs]ve t 7mx ny7_/7m n7 mu/ lie betwe5 x & y1 if x & y be unequal3 = if x be ! less ( ! two1 x is c]ta9ly great] ?an mx nx ----------- or ?an x2 m n & if y be ! great] ( ! two1 x is c]ta9ly less ?an my ny -----------1 or ?an y4 m n ,x !re=e lies betwe5 x & y4 ,n[ let a_/b be x1 & let c_/d be y3 !n a bx1 c dy4 ,n[ bx dy ----------- b d is some?+ betwe5 x & y1 z 0 j prov$2 !re=e a c --------- b d is some?+ betwe5 a_/b & c_/d4 ,aga91 s9ce a_/b & c_/d are respectively equal to ap_/bp & cq_/dq1 & s9ce1 z has j be5 prov$1 ap cq ----------- bp dq lies betwe5 ! two la/1 x also lies betwe5 ! two fir/2 t is1 if p & q be any numb]s or fractions :atsoev]1 ap cq ----------- bp dq lies betwe5 a_/b & c_/d4 #bja4 ,by ! la/ article we may (t5 =m "s notion ( ! value ( an expression too complicat$ to be easily calculat$4 ,?us1 #a x #a x #a -------- lies betwe5 --- & ----1 or #a & ---2 #a xx #a xx x ax by ax by -------------- lies betwe5 ----- & ------1 axx bbyy axx bbyy t is1 betwe5 #a_/x & #a_/;by4 ,& x has be5 %ewn t 7a b7_/#b lies betwe5 a & b1 ! d5om9ator be+ consid]$ z #a #a4 #bjb4 ,x may also be prov$ t a fraction su* z a b c d --------------------- p q r s a b c d always lies among ---1 ---1 ---1 & ---1 p q r s t is1 is less ?an ! greate/ ( !m1 & great] ?an ! lea/4 ,let ^! fractions be arrang$ 9 ord] ( magnitude2 t is1 let a_/p be great] ?an b_/q1 b_/q be great] ?an c_/r1 & c_/r great] ?an d_/s4 ,!n by 7#bjj7 is & less great] a b a ?an ?an b c --------- --- --- & --- p q p q r a b c a b a c d --------------- --------- & --- --- & --- p q r p q p r s a b c d a b c a d --------------------- --------------- & --- --- p q r s p q r p s :5ce ! proposition is evid5t4 #bjc4 ,x is usual to signify 8a is great] ?an b0 by a b & 8a is less ?an b0 by a b2 ! op5+ ( ,v be+ turn$ t[>ds ! great] quantity4 ,! pupil is recomm5d$ to make himself famili> ) ^! signs4 ,,section ,,ix4 ,,on ,,p]mutations ,,& ,,comb9ations4 #bjd4 ,if a numb] ( c\nt]s1 di/+ui%$ by diff]5t lett]s1 be plac$ on ! table1 & any numb] ( !m1 say f\r1 be tak5 away1 ! "q is1 to det]m9e 9 h[ _m diff]5t ways ?is c be done4 ,ea* way ( do+ x gives :at is call$ a comb9ation ( f\r1 b :i* might ) m propriety be call$ a selection ( f\r4 ,two comb9ations or selections are call$ diff]5t1 :i* diff] 9 any way :atev]2 ?us1 abcd & abce are diff]5t1 d be+ 9 "o & e 9 ! o!r1 ! rema9+ p>ts be+ ! same4 ,let !re be six c\nt]s1 a1 b1 c1 d1 e1 & f2 ! comb9ations ( ?ree :i* c be made out ( !m are tw5ty 9 numb]1 z foll[3 abc ace bcd bef abd acf bce cde abe ade bcf cdf abf adf bde cef acd aef bdf def ,! comb9ations ( f\r are fifte5 9 numb]1 namely1 abcd abde acde adef bcef abce abdf acdf bcde bdcf abcf abef acef bcdf cdef & s on4 #bje4 ,ea* ( ^! comb9ations may be writt5 9 sev]al diff]5t ord]s2 ?us1 abcd may be dispos$ 9 any ( ! foll[+ ways3 abcd acbd acdb abdc adbc adcb bacd cabd cadb badc dabc dacb bcad cbad cdab bdac dbac dcab bcda cbda cdba bdca dbca dcba ( :i* no two are entirely 9 ! same ord]4 ,ea* ( ^! is said to be a di/9ct p]mutation ( abcd4 ,consid]$ z a comb9ation1 !y are all ! same1 z ea* conta9s a1 b1 c1 & d4 #bjf4 ,we n[ proce$ to f9d h[ _m p]mutations1 ea* conta9+ "o giv5 numb]1 c be made f ! c\nt]s 9 ano!r giv5 numb]1 six1 = example4 ,if we knew h[ to f9d all ! p]mutations conta9+ f\r c\nt]s1 we might make ^? :i* conta9 five ?us3 ,take any "o :i* conta9s f\r1 = example1 abcf 9 :i* d & e are omitt$2 write d & e successively at ! end1 :i* gives abcfd1 abcfe1 & repeat ! same process ) e o!r p]mutation ( f\r2 ?us1 dabc gives dabce & dabcf4 ,no p]mutation ( five c escape u if we proce$ 9 ?is mann]1 provid$ only we "k ^? ( f\r2 = any giv5 p]mutation ( five1 z dbfea1 w arise 9 ! c\rse ( ! process f dbfe1 :i*1 accord+ to our rule1 furni%es dbfea4 ,nei!r w any p]mutation be repeat$ twice1 = dbfea1 if ! rule be foll[$1 c only arise f ! p]mutation dbfe4 ,if we beg9 9 ?is way to f9d ! p]mutations ( two out ( ! six1 a b c d e f ea* ( ^! gives five2 ?us1 a gives ab ac ad ae af b 444 ba bc bd be bf & ! :ole numb] is #f #e1 or #cj4 ,aga91 ab gives abc abd abe abf ac 444 acb acd ace acf & "h are #cj1 or #f #e p]mutations ( #b1 ea* ( :i* gives #d p]mutations ( #c2 ! :ole numb] ( ! la/ is !re=e #f #e #d1 or #abj4 ,aga91 abc gives abcd abce abcf abd 444 abdc abde abdf & "h are #abj1 or #f #e #d1 p]mutations ( ?ree1 ea* ( :i* gives #c p]mutations ( f\r2 ! :ole numb] ( ! la/ is !re=e #f #e #d #c1 or #cfj4 ,9 ! same way1 ! numb] ( p]mutations ( #e is #f #e #d #c #b1 & ! numb] ( p]mutations ( six1 or ! numb] ( diff]5t ways 9 :i* ! :ole six c be arrang$1 is #f #e #d #c #b #a4 ,! la/ two results are ! same1 :i* mu/ be2 = s9ce a p]mutation ( five only omits "o1 x c only furni% "o p]mutation ( six4 ,if in/ead ( six we *oose any o!r numb]1 x1 ! numb] ( p]mutations ( two w be x7x-#a71 t ( ?ree w be x7x-#a77x-#b71 t ( f\r x7x -#a77x-#b77x-#c71 ! rule be+3 ,multiply ! :ole numb] ( c\nt]s by ! next less numb]1 & ! result by ! next less1 & s on1 until z _m numb]s h be5 multipli$ toge!r z !re are to be c\nt]s 9 ea* p]mutation3 ! product w be ! :ole numb] ( p]mutations ( ! sort requir$4 ,?us1 out ( #ab c\nt]s1 p]mutations ( f\r may be made to ! numb] ( #ab #aa #aj #i1 or #aahhj4 ,,ex]cises4 #bjg4 ,9 h[ _m diff]5t ways c eight p]sons be arrang$ on eight seats8 ,answ]1 #djcbj4 ,9 h[ _m ways c eight p]sons be seat$ at a r\nd table1 s t all %all n h ! same neighb\rs 9 any two arrangem5ts8#cj ,answ]1 #ejdj4 #cj ,! diff]5ce betwe5 ?is problem & ! la/ is left to ! +5uity ( ! pupil4 ,if ! hundr$? "p ( a f>?+ be giv5 = e diff]5t arrangem5t :i* c be made ( fifte5 p]sons1 to h[ mu* w ! :ole am\nt8 ,answ]1 #acfbafjh4 ,out ( sev5te5 consonants & five v[els1 h[ _m words c be made1 hav+ two consonants & "o v[el 9 ea*8 ,answ]1 #djhj4 #bjh4 ,if two or m ( ! c\nt]s h ! same lett] ^u !m1 ! numb] ( di/9ct p]mutations is less ?an t giv5 by ! la/ rule4 ,let !re be a1 a1 a1 b1 c1 d1 &1 = a mom5t1 let u di/+ui% betwe5 ! ?ree z ?us1 a1 a1 a4 ,!n1 abcaad1 & abcaad are reckon$ z di/9ct p]mutations 9 ! rule1 :]eas !y w\ld n h be5 s1 _h x n be5 = ! acc5ts4 ,to compute ! numb] ( di/9ct p]mutations1 let u make "o ) b1 c1 & d1 leav+ places = ! as1 ?us1 7 7 bc 7 7 7 7 d4 ,if ! as _h be5 di/+ui%$ z a1 a1 a1 we might h made #c #b #a di/9ct p]mutations1 by fill+ up ! vacant places 9 ! above1 all :i* six are ! same :5 ! as are n di/+ui%$4 ,h5ce1 to d$uce ! numb] ( p]mutations ( a1 a1 a1 b1 c1 d1 f t ( aaabcd1 we mu/ divide ! latt] by #c #b #a1 or #f1 :i* gives #f #e #d #c #b #a --------------------- or #abj4 #c #b #a ,simil>ly1 ! numb] ( p]mutations ( aaaabbbcc is #i #h #g #f #e #d #c #b #a --------------------------------- #d #c #b #a #c #b #a #b #a4 ,,ex]cise4 ,h[ _m v>iations c be made ( ! ord] ( ! lett]s 9 ! ^w antitr9it>ian8 ,answ]1 #abfabfjjj4 #bji4 ,f ! numb] ( p]mutations we c easily d$uce ! numb] ( comb9ations4 ,b1 9 ord] to =m ^! comb9ations indep5d5tly1 we w %ew a me?od simil> to t 9 7#bjf74 ,if we "k ! comb9ations ( two :i* c be made out ( a1 b1 c1 d1 e1 we c f9d ! comb9ations ( ?ree1 by writ+ successively at ! end ( ea* comb9ation ( two1 ! lett]s :i* come aft] ! la/ conta9$ 9 x4 ,?us1 ab gives abc1 abd1 abe2 ad gives ade only4 ,no comb9ation ( ?ree c escape u if we proce$ 9 ?is mann]1 provid$ only we "k ! comb9ations ( two2 = any giv5 comb9ation ( ?ree1 z acd1 w arise 9 ! c\rse ( ! process f ac1 :i*1 accord+ to our rule1 furni%es acd4 ,nei!r w any comb9ation be repeat$ twice1 = acd1 if ! rule be foll[$1 c only arise f ac1 s9ce nei!r ad nor cd furni%es x4 ,if we beg9 9 ?is way to f9d ! comb9ations ( ! five1 a b c d e a gives ab ac ad ae b bc bd be c cd ce d de ,( ^!1 ab gives abc abd abe ac acd ace ad ade bc bcd bce bd bde cd cde ae be ce & de give none4 ,( ^!1 abc gives abcd abce abd abde acd acde bcd bcde ,^? :i* conta9 e give none1 z be=e4 ,( ! la/1 abcd gives abcde1 & ! o!rs none1 :i* is evid5tly true1 s9ce only "o selection ( five c be made out ( five ?+s4 #baj4 ,! rule = calculat+ ! numb] ( comb9ations is d]iv$ directly f t = ! numb] ( p]mutations4 ,take #g c\nt]s2 !n1 s9ce ! numb] ( p]mutations ( two is #g #f1 & s9ce two p]mutations1 ba & ab1 are 9 any comb9ation ab1 ! numb] ( comb9ations is half t ( ! p]mutations1 or 7#g #f7_/b4 ,s9ce ! numb] ( p]mutations ( ?ree is #g #f #e1 & z ea* comb9ation abc has #c #b #a p]mutations1 ! numb] ( comb9ations ( ?ree is #g #f #e ----------4 #a #b #c ,also1 s9ce any comb9ation ( f\r1 abcd1 conta9s #d #c #b #a p]mutations1 ! numb] ( comb9ations ( f\r is #g #f #e #d -------------1 #a #b #c #d & s on4 ,! rule is3 ,to f9d ! numb] ( comb9ations1 ea* conta9+ n c\nt]s1 divide ! correspond+ numb] ( p]mutations by ! product ( #a1 #b1 #c1 c4 up to n4 ,if x be ! :ole numb]1 ! numb] ( comb9ations ( two is x7x - #a7 -------------2 #a #b t ( ?ree is x7x - #a77x - #b7 ---------------------2 #a #b #c t ( f\r is x7x - #a77x - #b77x - #c7 ------------------------------2 & s on4 #a #b #c #d #baa4 ,! rule may 9 half ! cases be simplifi$1 z foll[s4 ,out ( t5 c\nt]s1 = e di/9ct selection ( sev5 :i* is tak51 a di/9ct comb9ation ( #c is left4 ,h5ce1 ! numb] ( comb9ations ( sev5 is z _m z t ( ?ree4 ,we may1 !re=e1 f9d ! comb9ations ( ?ree in/ead ( ^? ( sev52 & we mu/ moreov] expect1 & may ev5 ass]t1 t ! two =mul = f9d+ ^! two numb]s ( comb9ations are ! same 9 result1 ?\gh diff]5t 9 =m4 ,& s x proves2 = ! numb] ( comb9ations ( sev5 out ( t5 is #aj #i #h #g #f #e #d --------------------------1 #a #b #c #d #e #f #g 9 :i* ! product #g #f #e #d occurs 9 bo? t]ms1 & !re=e may be remov$ f bo? 7#ajh71 leav+ #aj #i #h ----------1 #a #b #c :i* is ! numb] ( comb9ations ( ?ree out ( t54 ,! same may be %ewn 9 o!r cases4 ,,ex]cises4 ,h[ _m comb9ations ( f\r c be made out ( twelve ?+s8 ,answ]1 #die4 ,:at numb] #f #h #bh ( comb9ations #d out ( #aa ,answ]1 #ccj c be made ( #bf #bh #cgh #f #ae #ejje ,h[ _m comb9ations c be made ( #ac out ( #eb2 or h[ _m diff]5t h&s may a p]son hold at ! game ( :i/8 ,answ]1 #fcejaceeifjj4 ,,book ,,ii4 ,,comm]cial ,,ari?metic4 ,,section ,i4 ,,weights1 ,,measures1 ,c4 #bab4 ,9 mak+ ! calculations :i* are necess>y 9 comm]cial affairs1 no m processes are requir$ ?an ^? :i* h be5 expla9$ 9 ! prec$+ book4 ,b !re is /ill "o ?+ want$--n to insure ! accuracy ( our calculations1 b to enable u to comp>e & judge ( _! results4 ,we h hi!rto made use ( a s+le unit 7#ae71 & h treat$ ( o!r quantities :i* are made up ( a numb] ( units1 9 ,sections ,,ii41 ,,iii41 & ,,iv41 & ( ^? :i* conta9 p>ts ( t unit 9 ,sections ,v4 & ,,vi4 ,?us1 if we are talk+ ( di/ances1 & take a mile z ! unit1 any o!r l5g? may be repres5t$1#ca ei!r by a c]ta9 numb] ( miles1 or a c]ta9 numb] ( p>ts ( a mile1 & 7#a mean+ "o mile7 may be express$ ei!r by a :ole numb] or a fraction4 ,b we c easily see t 9 _m cases inconv5i5ces w\ld arise4 ,suppose1 = example1 ,i say1 t ! l5g? ( "o room is _/ ( a mile1 & ( ano!r _/ ( a mile1 :at idea c we =m z to h[ mu* ! second is long] ?an ! fir/8 ,x is necess>y to h "s small] measure2 & if we divide a mile into #agfj equal p>ts1 & call ea* ( ^! p>ts a y>d1 we %all f9d t ! l5g? ( ! fir/ room is #i y>ds & _/ ( a y>d1 & t ( ! second #aj y>ds & _/ ( a y>d4 ,f ?is we =m a mu* bett] notion ( ^! diff]5t l5g?s1 b /ill n a v p]fect "o1 on acc\nt ( ! fractions _/ & _/4 ,to get a cle>] idea ( ^!1 suppose ! y>d to be divid$ into ?ree equal p>ts1 & ea* ( ^! p>ts to be call$ a foot2 !n _/ ( a y>d conta9s #b feet1 & _/ ( a y>d conta9s _/ ( a foot1 or a little m ?an ( a foot4 ,!re=e ! l5g? ( ! fir/ room is n[ #i y>ds1 #b feet1 & ( a foot2 t ( ! second is #aj y>ds & a little m ?an ( a foot4 ,we see1 !n1 ! conv5i5ce ( hav+ l>ge measures = l>ge quantities1 & small] measures = small ones2 b ?is is done = conv5i5ce only1 = x is possible to p]=m calculations ^u any sort ( quantity1 ) "o measure alone1 z c]ta9ly z ) m ?an "o2 & n only possible1 b m conv5i5t1 z f> z ! m]e calculation is conc]n$4 #ca ,x is n true1 t if we *oose any quantity z a unit1 any o!r quantity ( ! same k9d c be exactly repres5t$ ei!r by a c]ta9 numb] ( units1 or ( p>ts ( a unit4 ,to und]/& h[ ?is is prov$1 ! pupil w\ld require m k ?an he c be suppos$ to h2 b we c %ew him t1 = any ?+ he kn[s to ! contr>y1 !re may be quantities :i* are nei!r units nor p>ts ( ! unit4 ,take a ma!matical l9e ( "o foot 9 l5g?1 divide x into t5 p>ts1 ea* ( ^? p>ts into t5 p>ts1 & s on cont9ually4 ,if a po9t ,a be tak5 at haz>d 9 ! l9e1 x does n appe> self-evid5t t if ! decimal division be cont9u$ ev] s f>1 "o ( ! po9ts ( division mu/ at la/ fall exactly on ,a3 nei!r w\ld ! same appe> necess>ily true if ! division 7 made into sev5?s1 or elev5?s1 or 9 any o!r way4 ,!re may !n possibly be a "p ( a foot :i* is no exact num]ical fraction :atev] ( ! foot2 & ?is1 9 a high] bran* ( ma!matics1 is f\nd to be ! case times )\t numb]4 ,:at is meant 9 ! words on :i* ?is note is writt51 is1 t any "p ( a foot c be repres5t$ z ne>ly z we please by a num]ical fraction ( x2 & ?is is suffici5t = practical purposes4 ,! measures :i* are us$ 9 ?is c\ntry are n ^? :i* w\ld h be5 *os5 _h !y be5 made all at "o "t1 & by a p well acqua9t$ ) ari?metic & natural philosophy4 ,we proce$ to %ew h[ ! results ( ! latt] sci5ce are made useful 9 our sy/em ( measures4 ,:e!r ! circum/ances introduc$ are suffici5tly well kn[n to r5d] ! foll[+ me?ods exact en\gh = ! recov]y ( a/ronomical /&>ds1 may be matt] ( op9ion2 b no d\bt c be ent]ta9$ ( _! be+ amply correct = comm]cial purposes4 ,x is evid5tly desirable t weights & measures %\ld always cont9ue ! same1 & t po/]ity %\ld be able to replace any "o ( !m :5 ! orig9al measure is lo/4 ,x is true t a y>d1 :i* is n[ exact1 is kept by ! public au?orities2 b if ?is 7 burnt by accid5t1#cb h[ are ^? :o %all live #ejj ye>s h5ce to "k :at 0 ! l5g? :i* _! ance/ors call$ a y>d8 ,to ensure !m ?is k1 ! measure mu/ be d]iv$ f some?+ :i* _c be alt]$ by man1 ei!r f design or accid5t4 ,we f9d su* a quantity 9 ! "t ( ! daily revolution ( ! e>?1 & also 9 ! l5g? ( ! ye>1 bo? ( :i*1 z is %ewn 9 a/ronomy1 w rema9 ! same1 at lea/ = an enorm\s numb] ( c5turies1 unless "s great & totally unkn[n *ange take place 9 ! sol> sy/em4 ,s long z a/ronomy is cultivat$1 x is impossible to suppose t ei!r ( ^! w be lo/1 & x is kn[n t ! latt] is #cfebdbbd mean sol> days1 or ab\t #cfe ( ! av]age int]val :i* elapses betwe5 noon & noon1 t is1 betwe5 ! times :5 ! sun is highe/ 9 ! heav5s4 ,our ye> is made to consi/ ( #cfe days1 & ! odd qu>t] is all[$ = by add+ "o day to e f\r? ye>1 :i* gives :at we call leap-ye>4 ,?is is ! same z add+ ( a day to ea* ye>1 & is r too mu*1 s9ce ! excess ( ! ye> above #cfe days is n #be b #bdbbd ( a day4 ,! diff]5ce is #jjggf ( a day1 :i* is ! quantity by :i* our av]age ye> is too long4 ,?is am\nts to a day 9 ab\t #abh ye>s1 or to ab\t #c days 9 #d c5turies4 ,! error is correct$ by all[+ only "o out ( f\r ( ! ye>s :i* close ! c5turies to be leap-ye>s4 ,?us1 ,a4,d4 #ahjj & #aijj are n leap-ye>s1 b #bjjj is s4 #cb ,s9ce ?is 0 fir/ writt51 ! accid5t has happ5$4 ,! /&>d y>d 0 s injur$ z to be r5d]$ useless by ! fire at ! ,h\ses ( ,p>liam5t4 #bac4 ,! day is !re=e ! fir/ measure obta9$1 & is divid$ into #bd p>ts or h\rs1 ea* ( :i* is divid$ into #fj p>ts or m9utes1 & ea* ( ^! aga9 into #fj p>ts or seconds4 ,"o second1 m>k$ ?us1 #a1cc is !re=e ! #hfdjj "p ( a day1 & ! foll[+ is ! ,,measure ,,( ,,"t4#cd #fj seconds are #a m9ute #a m4 #fj m9utes 0 #a h\r #a h4 #bd h\rs 0 #a day #a d4 #g days 0 #a week #a wk4 #cfe days 0 #a ye> #a yr4 #bad4 ,! second hav+ be5 obta9$1 a p5dulum c be con/ruct$ :i* %all1 :5 put 9 motion1 p]=m "o vibration 9 exactly "o second1 9 ! latitude ( ,gre5wi*4#ce ,if we 7 inv5t+ measures1 x w\ld be conv5i5t to call ! l5g? ( ?is p5dulum a y>d1 & make x ! /&>d ( all our measures ( l5g?4 ,b z !re is a y>d already e/abli%$1 x w d equally well to tell ! l5g? ( ! p5dulum 9 y>ds4 ,x 0 f\nd by commission]s appo9t$ = ! purpose1 t ?is p5dulum 9 ,london 0 #ciacic in*es1 or ab\t "o y>d1 ?ree in*es1 & _/ ( an in*4 ,! foll[+ is ! division ( ! y>d4 ,,measures ,,( ,,l5g?4 ,! l[e/ measure is a b>leycorn4#cf #c b>leycorns are #a in* #a 94 #ab in*es #a foot #a ft4 #c feet #a y>d #a yd4 #e y>ds #a pole #a po4 #dj poles or #bbj y>ds #a furlong #a fur4 #h furlongs or #agfj y>ds #a mile #a mi4 ,also #f feet #a fa?om #a f?4 #fi miles #a degree #a deg4 or #a4 #cc ,! m9ute & second are (t5 m>k$ ?us1 #a1 #a3 b ?is notation is n[ almo/ entirely appropriat$ to ! m9ute & second ( angul> measure4 #cd ,! measures 9 italics are ^? :i* x is mo/ necess>y t ! /ud5t %\ld le>n by he>t4 #ce ,! l5g?s ( ! p5dulums :i* w vibrate 9 "o second are slightly diff]5t 9 diff]5t latitudes4 ,gre5wi* is *os5 z ! /ation ( ! ,royal ,obs]vatory4 ,we may add1 t mu* d\bt is n[ ent]ta9$ z to ! sy/em ( /&>ds d]iv$ f nature be+ capable ( t extreme accuracy :i* 0 once attribut$ to x4 #cf ,! in* is said to h be5 orig9ally obta9$ by putt+ toge!r ?ree gra9s ( b>ley4 ,a geographical mile is _/? ( a degree1 & ?ree su* miles are "o nautical league4 ,9 ! measurem5t ( clo? or l95 ! foll[+ are also us$3 #b in*es are #a nail #a nl4 #d nails #a qu>t] 7( a y>d7 #a qr4 #c qu>t]s #a ,flemi% ell #a ,fl4 e4 #e qu>t]s #a ,engli% ell #a ,e4 e4 #f qu>t]s #a ,fr5* ell #a ,fr4 e4 #bae4 ,,measures ,,( ,,surface1 ,,or ,,sup]ficies4 ,all surfaces are measur$ by squ>e in*es1 squ>e feet1 c42 ! squ>e in* be+ a squ>e ^: side is an in* 9 l5g?1 & s on4 ,! foll[+ measures may be d$uc$ f ! la/1 z w aft]w>ds appe>4 #add squ>e in*es are #a squ>e foot #a sq4 ft4 #i squ>e feet #a squ>e y>d #a sq4 yd4 #cj squ>e y>ds #a squ>e pole #a sq4 p4 #dj squ>e poles #a rood #a rd4 #d roods #a acre #a ac4 ,?us1 ! acre conta9s #dhdj squ>e y>ds1 :i* is t5 times a squ>e ( #bb y>ds 9 l5g? & bread?4 ,?is #bb y>ds is ! l5g? :i* l&-surveyors' *a9s are made to h1 & ! *a9 is divid$ into #ajj l9ks1 ea* #bb ( a y>d or #gib in*es4 ,an acre is !n #aj squ>e *a9s4 ,x may also be notic$ t a squ>e ^: side is #fi_/ y>ds is ne>ly an acre1 n exce$+ x by ( a squ>e foot4 #baf4 ,,measures ,,( ,,solidity ,,or ,,capacity4#cg ,cubes are solids hav+ ! figure ( dice4 ,a cubic in* is a cube ea* ( ^: sides is an in*1 & s on4 #agbh cubic in*es are #a cubic foot #a c4 ft4 #bg cubic feet #a cubic y>d #a c4 yd4 #cg #8,capacity' is a t]m :i* _c be bett] expla9$ ?an by its use4 ,:5 "o measure holds m ?an ano!r1 x is said to be m capaci\s1 or to h a great] capacity4 ,?is measure is n mu* us$1 except 9 purely ma!matical que/ions4 ,9 ! measurem5ts ( diff]5t commodities v>i\s measures 7 us$1 :i* are n[ r$uc$1 by act ( p>liam5t1 to "o4 ,?is is commonly call$ ! imp]ial measure1 & is z foll[s3 ,,measure ,,( ,,liquids ,,& ,,( ,,all ,,dry ,,goods4 #d gills are #a p9t #a pt4 #b p9ts #a qu>t #a qt4 #d qu>ts #a gallon #a gall4 #b gallons #a peck#ch #a pk4 #d pecks #a bu%el #a bu4 #h bu%els #a qu>t] #a qr4 #e qu>t]s #a load #a ld4 ,! gallon 9 ?is measure is ab\t #bggbgd cubic in*es2 t is1 v ne>ly #bgg cubic in*es4#ci #bag4 ,! smalle/ weight 9 use is ! gra91 :i* is ?us det]m9$4 ,a vessel ^: int]ior is a cubic in*1 :5 fill$ ) wat]1#dj has its weight increas$ by #bebdeh gra9s4 ,( ! gra9s s det]m9$1 #gjjj are a p\nd av]dupois1 & #egfj a p\nd troy4 ,! fir/ p\nd is always us$1 except 9 weigh+ preci\s metals & /ones1 & also m$ic9es4 ,x is divid$ z foll[s3 #ch ,?is measure1 & ^? :i* foll[1 are us$ = dry goods only4 #ci ,s9ce ! publication ( ! ?ird edition1 ! heap$ measure1 :i* 0 "p ( ! new sy/em1 has be5 aboli%$4 ,! foll[+ p>agraph f ! ?ird edition w s]ve = ref]5ce to x3 8,! o!r imp]ial measure is appli$ to goods :i* x is cu/om>y to sell by heap$ measure1 & is z foll[s3 #b gallons #a peck #d pecks #a bu%el #c bu%els #a sack #ab sacks #a *aldron4 ,! gallon & bu%el 9 ?is measure hold ! same :5 only j fill$1 z 9 ! la/4 ,! bu%el1 h[ev]1 heap$ up z direct$ by ! act ( p>liam5t1 is a little m ?an "o-f\r? great] ?an be=e40 #dj ,pure wat]1 cle>$ f =eign sub/ances by di/illation1 at a temp]ature ( #fb ,fahr4 ,,av]dupois ,,weight4 #bg_/ gra9s are #a dram #a dr4 #f drams1 or dra*ms #a ounce#da #a oz4 #af ounces #a p\nd #a lb4 #bh p\nds #a qu>t] #a qr4 #d qu>t]s #a hundr$-weight #a cwt4 #bj hundr$-weight #a ton #a ton4 #da ,x is m common to divide ! ounce into f\r qu>t]s ?an into sixte5 drams4 ,! p\nd av]dupois conta9s #gjjj gra9s4 ,a cubic foot ( wat] weighs #fbcbajfjf p\nds av]dupois1 or #iigacfifia ounces4 ,= ! preci\s metals & = m$ic9es1 ! p\nd troy1 conta9+ #egfj gra9s1 is us$1 b is diff]5tly divid$ 9 ! two cases4 ,! measures are z foll[3 ,,troy ,,weight4 #bd gra9s are #a p5nyweight #a dwt4 #bj p5nyweights #a ounce #a oz4 #ab ounces #a p\nd #a lb4 ,! p\nd troy conta9s #egfj gra9s4 ,a cubic foot ( wat] weighs #gegcgd p\nds troy1 or #ijhhdhh ounces4 ,,apo!c>ies' ,,weight4 #bj gra9s are #a scruple #c scruples #a dram #h drams #a ounce #ab ounces #a p\nd lb #bah4 ,! /&>d co9s ( copp]1 silv]1 & gold1 are1--! p5ny1 :i* is #aj drams ( copp]2 ! %ill+1 :i* weighs #c p5nyweights #ae gra9s1 ( :i* #c p>ts out ( #dj are alloy1 & ! re/ pure silv]2 & ! sov]eign1 weigh+ #e p5nyweights & #c gra9s1 ( :i* #a "p out ( #ab is copp]1 & ! re/ pure gold4 ,,measures ,,( ,,money4 ,! l[e/ co9 is a f>?+1 :i* is m>k$ ?us1 1 be+ "o f\r? ( a p5ny4 #b f>?+s are #a halfp5ny d4 #b halfp5ce #a p5ny #a;d4 #ab p5ce #a %ill+ #as4 #bj %ill+s #a p\nd#db or sov]eign #a #ba %ill+s #a gu9ea4#dc #bai4 ,:5 any quantity is made up ( sev]al o!rs1 express$ 9 diff]5t units1 su* z #a4 #ad4 #f1 or #b;cwt4 #aqr4 #clbs41 x is call$ a comp\nd quantity4 ,f ^! tables x is evid5t t any comp\nd quantity ( any sub/ance c be measur$ 9 sev]al diff]5t ways4 ,= example1 ! sum ( money :i* we call five p\nds f\r %ill+s is also #ajd %ill+s1 or #abdh p5ce1 or #diib f>?+s4 ,x is easy to r$uce any quantity f "o ( ^! measurem5ts to ano!r2 & ! foll[+ examples w be suffici5t to %ew h[ to apply ! same process1 usually call$ ,,r$uction1 to all sorts ( quantities4 ,i4 ,h[ _m f>?+s are !re 9 #ah4 #ab4 #f8dd #db ,! ,engli% p\nd is g5]ally call$ a p\nd /]l+1 :i* di/+ui%es x f ! weight call$ a p\nd1 & also f =eign co9s4 #dc ,! co9 call$ a gu9ea is n[ no long] 9 use1 b ! "n is /ill giv51 f cu/om1 to #ba %ill+s4 ,! p\nd1 :i* 0 n a co91 b a note promis+ to pay #bj %ill+s to ! be>]1 is also disus$ = ! pres5t1 & ! sov]eign supplies its place2 b ! "n p\nd is /ill giv5 to #bj %ill+s4 #dd ,f>?+s are nev] writt5 b z p>ts ( a p5ny4 ,?us1 ?ree f>?+s be+ #c_/d ( a p5ny1 is writt5 #c_/d1 or 4 ,"o halfp5ny may be writt5 ei!r z #b_/d or 2 ! latt] is mo/ common4 ,s9ce !re are #bj %ill+s 9 a p\nd1 !re are1 9 #ah1 #ah #bj1 or #cfj %ill+s2 !re=e1 #ah4 #ab is #cfj #ab1 or #cgb %ill+s4 ,s9ce !re are #ab p5ce 9 a %ill+1 9 #cgb %ill+s !re are #cgb #ab1 or #ddfd p5ce2 &1 !re=e1 9 #ah4 #ab4 #f !re are #ddfd #f1 or #ddgj p5ce4 ,s9ce !re are #d f>?+s 9 a p5ny1 9 #ddgj p5ce !re are #ddgj #d1 or #aghhj f>?+s2 &1 !re=e1 9 #ah4 #ab4 #f !re are #aghhj #c1 or #aghhc f>?+s4 ,! :ole ( ?is process may be writt5 z foll[s3 #ah 4 #ab 4 #f #bj -------- #cfj #ab #cgb #ab ----- #ddfd #f #ddgj #d ----- #aghhj #c #aghhc ,,ii4 ,9 #aghhc f>?+s1 h[ _m p\nds1 %ill+s1 p5ce1 & f>?+s are !re8 ,s9ce #aghhc1 divid$ by #d1 gives ! quoti5t #ddgj1 & ! rema9d] #c1 #aghhc f>?+s are #ddgj p5ce & #c f>?+s 7#bah74 ,s9ce #ddgj1 divid$ by #ab1 gives ! quoti5t #cgb1 & ! rema9d] #f1 #ddgj p5ce is #cgb %ill+s & #f p5ce4 ,s9ce #cgb1 divid$ by #bj1 gives ! quoti5t #ah1 & ! rema9d] #ab1 #cgb %ill+s is #ah p\nds & #ab %ill+s4 ,!re=e1 #aghhc f>?+s is #ddgj;d41 :i* is #cgbs4 #f;d41 :i* is #ah4 #ab4 #f4 ,! process may be writt5 z foll[s3 #d7aghhc ----- #ab7ddgj 444 #c ---- #bj7cgb 444 #f #ah 4 #ab 4 #f ,,ex]cises4 ,a has #ajj4 #d4 #aa1 & ,b has #fdcib f>?+s4 ,if ,a receive #adib f>?+s1 & ,b #a4 #b4 #c1 :i* w !n h ! mo/1 & by h[ mu*8--,answ]1 ,a w h #cc4 #ab4 #c m ?an ,b4 ,9 ! foll[+ table ! quantities writt5 opposite to ea* o!r are ! same3 ea* l9e furni%es two ex]cises4 #ae 4 #ah 4 #i #aecjb f>?+s4 #aae #a #h #ffcjgb gra9s4 #c #ad #i #ajja drams4 #c #adi yds #b #i 9 #aiedgg in*es4 #ai #b #a gall #b q #abfj p9ts4 #af #bc #dg #eijbg seconds4 #bbj4 ,! same may be done :]e ! numb] fir/ express$ is fractional4 ,= example1 h[ _m %ill+s & p5ce are !re 9 _/ ( a p\nd8 ,n[1 _/ ( a p\nd is _/ ( #bj %ill+s2 _/ ( #bj is #d #bj #d #d #af ------1 or ----- 7#aaj71 or ---1 #ae #c #c or 7#aje7 #e ( a %ill+4 ,aga91 ( a %ill+ is ( #ab p5ce1 or #d p5ce4 ,!re=e1 _/ #es4 #d;d4 ,also1 #bc ( a day is #bc #bd 9 h\rs1 or #eeb2 & #eb ( an h\r is #eb #fj 9 m9utes1 or #cb2 & #b ( a m9ute is #b #fj 9 seconds1 or #ab2 :5ce #bc ( a day is #e #ca #ab4 ,aga91 suppose x requir$ to f9d :at "p ( a p\nd #fs4 #h;d4 is4 ,s9ce #fs4 #h;d4 is #hj p5ce1 & s9ce ! :ole p\nd conta9s #bj #ab or #bdj p5ce1 #fs4 #h;d4 is made by divid+ ! p\nd into #bdj p>ts1 & tak+ #hj ( !m4 ,x is !re=e _/ 7#ajg71 b _/ 7#ajh72 !re=e1 #fs4 #h;d4 4 ,,ex]cises4 ( a day is #i #cf #abhda ( a day #c #d #edfbdde #beg ( a cwt4 #bh #ab #hgjd #adicf #b #aa #cchef #de ,:5 a decimal foll[s a :ole numb]1 ! decimal is always ( ! same unit z ! :ole numb]4 ,?us1 #ee is five seconds & five-t5?s ( a second4 ,?us1 #je means five-t5?s ( a second2 #jc1 ?ree-t5?s ( an h\r4 #bba1 #bbb4 ,i h ?\ght x be/ to ref] ! mode ( conv]t+ %ill+s1 p5ce1 & f>?+s into decimals ( a p\nd to ! ,app5dix 7,see ,app5dix ,on ,decimal ,money74 ,i %\ld /rongly recomm5d ! read] to make himself p]fectly famili> ) ! modes giv5 9 t ,app5dix4 ,to prev5t ! subsequ5t sections f be+ alt]$ 9 _! numb]+1 ,i h numb]$ ?is p>agraph z above4 #bbc4 ,! rule ( addition#df ( two comp\nd quantities ( ! same sort w be evid5t f ! foll[+ example4 ,suppose x requir$ to add #aib4 #ad4 #b to #fd4 #ac4 #aa4 ,! sum ( ^! two is ! :ole ( t :i* arises f add+ _! sev]al p>ts4 ,n[ d4 d4 _/d4 #j 4 #j 4 #a 7#bai7 #aa;d4 #b;d4 #ac;d4 #j 4 #a 4 #a #acs4 #ads4 #bgs4 #a 4 #g 4 #j #fd #aib #bef 4 #j 4 #j ----------- ,! sum ( all ( :i* is #beg4 #h 4 #b ,?is may be done at once1 & writt5 z foll[s3 #aib 4 #ad 4 #b #fd 4 #ac 4 #aa ---------------- #beg 4 #h 4 #b #df ,be=e read+ ?is article & ! next1 articles 7#bi7 & 7#db7 %\ld be read aga9 c>efully4 ,beg9 by add+ toge!r ! f>?+s1 & r$uce ! result to p5ce & f>?+s4 ,set d[n ! la/ only1 c>ry ! fir/ to ! l9e ( p5ce1 & add ! p5ce 9 bo? l9es to x4 ,r$uce ! sum to %ill+s & p5ce2 set d[n ! la/ only1 & c>ry ! fir/ to ! l9e ( %ill+s1 & s on4 ,! same me?od mu/ be foll[$ :5 ! quantities are ( any o!r sort2 & if ! tables be kept 9 memory1 ! process w be easy4 #bbd4 ,,subtraction is p]=m$ on ! same pr9ciple z 9 7#dj71 namely1 t ! diff]5ce ( two quantities is n alt]$ by add+ ! same quantity to bo?4 ,suppose x requir$ to subtract #ai 4 #ac4 #aj f #bd4 #e4 #g4 ,write ^! quantities "u "o ano!r ?us3 #bd4 #e4 #g #ai4 #ac4 #aj ,s9ce _c be tak5 f or _/1 add #a;d4 to bo? quantities1 :i* w n alt] _! diff]5ce2 or1 :i* is ! same ?+1 add #d f>?+s to ! fir/1 & #a;d4 to ! second4 ,! p5ce & f>?+s 9 ! two l9es !n /& ?us3 #g_/;d4 & #aa;d4 ,n[ subtract f _/1 & ! diff]5ce is :i* mu/ be writt5 "u ! f>?+s4 ,aga91 s9ce #aa;d4 _c be subtract$ f #g;d41 add #as4 to bo? quantities by add+ #ab;d4 to ! fir/1 & #as4 to ! second4 ,! p5ce 9 ! fir/ l9e are !n #ai1 & 9 ! second #aa1 & ! diff]5ce is #h1 :i* write "u ! p5ce4 ,s9ce ! %ill+s 9 ! l[] l9e 7 increas$ by #a1 !re are n[ #ads4 9 ! l[]1 & #es4 9 ! upp] "o4 ,add #bjs4 to ! upp] & #a to ! l[] l9e1 & ! subtraction ( ! %ill+s 9 ! second f ^? 9 ! fir/ leaves #aas4 ,aga91 !re are n[ #bj 9 ! l[]1 & #bd 9 ! upp] l9e1 ! diff]5ce ( :i* is #d2 !re=e ! :ole diff]5ce ( ! two sums is #d4 #aa4 #h4 ,if we write d[n ! two sums ) all ! additions :i* h be5 made1 ! process w /& ?us3 #bd 4 #be 4 #ai_/ #bj 4 #ad 4 #aa ------------------ ,diff]5ce #d 4 #aa 4 #h #bbe4 ,! same me?od may be appli$ to any ( ! quantities 9 ! tables4 ,! foll[+ is ano!r example3 ,f #g cwt4 #b qrs4 #ba lbs4 #ad oz4 ,subtract #b cwt4 #c qrs4 #bg lbs4 #ab oz4 ,aft] alt]ations h be5 made simil> to ^? 9 ! la/ article1 ! "q becomes3 ,f #g cwt4 #f qrs4 #di lbs4 #ad oz4 ,subtract #c cwt4 #d qrs4 #bg lbs4 #ab oz4 ---------------------------- ,! diff]5ce is #d cwt4 #b qrs4 #bb lbs4 #b oz4 ,9 ?is example1 & almo/ e o!r1 ! process may be a little %ort5$ 9 ! foll[+ way4 ,"h we d n subtract #bg lbs4 f #ba lbs41 :i* is impossible1 b we increase #ba lbs4 by #a qr4 or #bh lbs4 & !n subtract #bg lbs4 f ! sum4 ,x w\ld be %ort]1 & lead to ! same result1 fir/ to subtract #bg lbs4 f #a qr4 or #bh lbs4 & add ! diff]5ce to #ba lbs4 #bbf4 ,,ex]cises4 ,a man has ! foll[+ sums to receive3 #aic4 #ad4 #aa1 #bb4 #j4 #f1 #fdgc4 #j4 #j1 & #di4 #ad4 #d2 & ! foll[+ debts to pay3 #bjj 4 #ai4 #f1 #cje4 #af4 #aa1 #bb1 & #ai4 #f4 #j4 ,h[ mu* w rema9 aft] pay+ ! debts8 ,answ]1 #faij4 #g4 #d4 ,!re are f\r t[ns1 9 ! ord] ,a1 ,b1 ,c1 & ,d4 ,if a man c g f ,a to ,b 9 #e #bj #cc1 f ,b to ,c 9 #f #di #b & f ,a to ,d 9 #ai #j #ag1 h[ long w he be 9 go+ f ,b to ,d1 & f ,c to ,d8 ,answ]1 #ac #ci #dd1 & #f #ej #db4 #bbg4 ,9 ord] to p]=m ! process ( ,,multiplication1 x mu/ be recollect$ t1 z 9 7#eb71 if a quantity be divid$ into sev]al p>ts1 & ea* ( ^! p>ts be multipli$ by a numb]1 & ! products be add$1 ! result is ! same z w\ld arise f multiply+ ! :ole quantity by t numb]4 ,x is requir$ to multiply #g4 #ac4 #f by #ac4 ,! fir/ quantity is made up ( #g p\nds1 #ac %ill+s1 #f p5ce1 & #a f>?+4 ,& #a f>?4 #ac is #ac f>?4 or #j 4 #j 4 #c 7#bai7 #f p5ce #ac is #gh p5ce1 or #j 4 #f 4 #f #ac %ill4 #ac is #afi %ill4 or #h 4 #i 4 #j #g p\nds #ac is #ia p\nds1 or #ia 4 #j 4 #j -------------- ,! sum ( all ^! is #ii 4 #ae 4 #i :i* is !re=e #g4 #ac4 #f #ac4 ,?is process is usually writt5 z foll[s3 #g 4 #ac 4 #f #ac ------------- #ii 4 #ae 4 #i #bbh4 ,,division is p]=m$ ^u ! same pr9ciple z 9 7#gd71 viz4 t if a quantity be divid$ into any numb] ( p>ts1 & ea* "p be divid$ by any numb]1 ! diff]5t quoti5ts add$ toge!r w make up ! quoti5t ( ! :ole quantity divid$ by t numb]4 ,suppose x requir$ to divide #ii4 #ae4 #i by #ac4 ,s9ce #ii divid$ by #ac gives ! quoti5t #g1 & ! rema9d] #h1 ! quantity is made up ( #ac #g1 or #ia1 & #h4 #ae4 #i4 ,! quoti5t ( ! fir/1 #ac be+ ! divisor1 is #g3 x rema9s to f9d t ( ! second4 ,s9ce #h is #afjs41 #h4 #ae4 #i is #ages4 #i;d41 & #age divid$ by #ac gives ! quoti5t #ac1 & ! rema9d] #f2 t is1 #ages4 #i;d4 is made up ( #afis4 & #fs4 #i;d41 ! quoti5t ( ! fir/ ( :i* is #acs41 & x rema9s to f9d t ( ! second4 ,s9ce #fs4 is #gb;d41 #fs4 #i;d4 is #ha;d41 & #ha divid$ by #ac gives ! quoti5t #f & rema9d] #c2 t is1 #ha;d4 is #gh;d4 & #c;d41 ( ! fir/ ( :i* ! quoti5t is #f;d4 ,aga91 s9ce #c;d4 is _/1 or #ab f>?+s1 #c;d4 is #ac f>?+s1 ! quoti5t ( :i* is #a f>?+1 or 1 )\t rema9d]4 ,we h !n divid$ #ii4 #ae4 #i into f\r p>ts1 ea* ( :i* is divisible by #ac1 viz4 #ia1 #afis41 #gh;d41 & #ac f>?+s2 s t ! ?irte5? "p ( ?is quantity is #g4 #ac4 #f4 ,! :ole process may be writt5 d[n z foll[s2 & ! same sort ( process may be appli$ to ! ex]cises :i* foll[3 s4 d4 s4 d4 #ac7ii #ae #i7 #g #ac #f #ia -- #h #bj --- #afj #ae #age #ac --- #de #ci -- #f #ab -- #gb #i #ha #gh -- #c #d -- #ab #a #ac #ac -- #j ,"h1 ea* ( ! numb]s #ii1 #age1 #ha1 & #ac1 is divid$ by #ac 9 ! usual way1 ?\gh ! divisor is only writt5 be=e ! fir/ ( !m4 ,,ex]cises4 #b cwt4 #a qr4 #ba lbs4 #g oz4 #ec #abi cwt4 #a qr4 #af lbs4 #c oz4 #b #d #c #bg #aji #bcf #aj #af #c #bg 4 #aj 4 #h #efi #aefff 4 #i 4 #d #g 4 #d 4 #h #abc #hhi 4 #ad #aff _/ #dj 4 #d 4 #aj_/ #ahg 4 #f 4 #g _/ #e 4 #ab 4 #d _/ #ds4 #f;d4 #aaba #bed 4 #aa 4 #b #ds4 #d;d4 #dbfj #fs4 #f;d4 #bhdj #bbi4 ,suppose x requir$ to f9d h[ _m times #as4 #d;d4 is conta9$ 9 #c4 #ai4 #aj4 ,! way to d ?is is to f9d ! numb] ( f>?+s 9 ea*4 ,by #bai1 9 ! fir/ !re are #fe1 & 9 ! second #chce f>?+s4 ,n[1 #chce conta9s #fe #ei times2 & !re=e ! second quantity is #ei times z great z ! fir/4 ,9 ! case1 h[ev]1 ( p\nds1 %ill+s1 & p5ce1 x w\ld be be/ to use decimals ( a p\nd1 :i* w give a suffici5tly exact answ]4 ,?us #as4 #d;d4 is #jfg1 & #c4 #ai4 #aj is #ciid1 & #ciid divid$ by #jfg is #ciid by #fg1 or #ei_/4 ,?is is an extreme case1 = ! small] ! divisor1 ! great] ! effect ( an error 9 a giv5 place ( decimals4 ,,ex]cises4 ,h[ _m times does #f cwt4 #b qrs4 conta9 #a qr4 #ad lbs4 #a oz48 & #a #b #j #dg conta9 #c #df8 ,answ]1 #agcjgeh & #dadcfgbeg4 ,if #b cwt4 #c qrs4 #a lb4 co/ #aej4 #ac4 #aj1 h[ mu* does #a lb4 co/8 ,answ]1 #is4 #i;d4 _/4 ,a groc] mixes #b cwt4 #ae lbs4 ( sug> at #aa;d4 p] p\nd ) #ad cwt4 #c lbs4 at #e;d4 p] p\nd4 ,at h[ mu* p] p\nd mu/ he sell ! mixture s z n to lose by mix+ !m8 ,answ]1 #e;d4 _/4 #bcj4 ,!re is a conv5i5t me?od ( multiplication call$ ,,practice4 ,suppose ,i ask1 ,h[ mu* d #aec tons co/ if ea* ton co/ #b4 #ae4 #g8 ,x is pla9 t if ?is sum be multipli$ by #aec1 ! product is ! price ( ! :ole4 ,b ?is is also evid5t1 t1 if ,i buy #aec tons at #b4 #ae4 #g ea* ton1 paym5t may be made by fir/ putt+ d[n #b = ea* ton1 !n #ajs4 = ea*1 !n #es41 !n #f;d41 & !n #a;d4 ,^! sums toge!r make up #b4 #ae4 #g1 & ! reason = ?is sep>ation ( #b4 #ae 4 #g into diff]5t p>ts w be soon app>5t4 ,! process may be c>ri$ on z foll[s3 #a4 #aec tons1 at #b ea* ton1 w co/ #cjf #j #j #b4 ,s9ce #ajs4 is 1 #aec tons1 at #ajs4 ea*1 w co/ #ae_/1 :i* is #gf #aj #j #c4 ,s9ce #es4 is ( #ajs41 #aec tons1 at #es41 w co/ half z mu* z ! same numb] at #ajs4 ea*1 t is1 ( #gf 4 #aj1 :i* is #ch #e #j #d4 ,s9ce #f;d4 is ( #es41 #aec tons1 at #f;d4 ea*1 w co/ ( :at ! same numb] co/s at #es4ea*1 t is1 ( #ch 4 #e1 :i* is #c #af #f #e4 ,s9ce #a or #c halfp5ce is ( #f;d4 or #ab halfp5ce1 #aec tons1 at #a;d4 ea*1 w co/ ( :at ! same numb] co/s at #f;d4 ea*1 t is1 (#c 4 #af 4 #f1 :i* is #j #ai #a ----------- ,! sum ( all ^! quantities is #dbe #aj #g :i* is1 !re=e1 #b 4 #ae 4 #g #aec4 ,! :ole process may be writt5 d[n z foll[s3 or :at #aec tons w\ld #aec #j #j co/ at #a p] ton4 ----------- ----------- #b is #b #a #cjf #j #j #b #j #j #ajs4 is ( #a #gf #aj #j #j #aj #j #es4 is ( #ajs4 #ch #e #j #j #e #j #f;d4 is ( #es4 #c #af #f #j #j #f #a;d4 is ( #f;d4 #j #ai #a #j #j #a ------------ ---------- ,sum #dbe #aj #g #b #ae #g ,,ano!r ,,example4 ,:at d #agce lbs4 co/ at #is4 #aj;d4 p] lb48 ,! price #is4 #aj;d4 is made up ( #es41 #ds41 #aj;d41 d41 & d42 ( :i* #es4 is ( #a1 #ds4 is ( #a1 #aj;d4 is ( #es41 d4 is _/ ( #aj;d41 & d4 is ( d4 ,foll[ ! same me?od z 9 ! la/ example1 :i* gives ! foll[+3 or :at #agce tons w\ld #agce #j #j co/ at #a p] lb4 ------------ ----------- #es4 is ( #a #dcc #ae #j #j #e #j #ds4 is ( #a #cdg #j #j #j #d #j #aj;d4 is ( #es4 #gb #e #aj #j #j #aj d4 is _/ ( #aj;d4 #c #ab #c #j #j #j d4 is ( d4 #a #af #a #j #j #j ------------- ------------ by addition 444 #heh #i #c #j #i #aj ,9 all cases1 ! price mu/ fir/ be divid$ into a numb] ( p>ts1 ea* ( :i* is a simple fraction#dg ( "s "o :i* goes be=e4 ,no rule c be giv5 = do+ ?is1 b practice w enable ! /ud5t imm$iately to f9d out ! be/ me?od = ea* case4 ,:5 t is done1 he mu/ f9d h[ mu* ! :ole quantity w\ld co/ if ea* ( ^! p>ts 7 ! price1 & !n add ! results toge!r4 #dg ,any fraction ( a unit1 ^: num]ator is unity1 is g5]ally call$ an aliquot "p ( t unit4 ,?us1 #bs4 & #ajs4 are bo? aliquot p>ts ( a p\nd1 be+ & 4 ,,ex]cises4 ,:at is ! co/ ( #bdc cwt4 at #ad 4 #ah 4 #h p] cwt48--,answ]1 #cfbi 4 #a 4 #j4 #afi bu%els at #b 4 #a 4 #c p] bu%el8--,answ]1 #cdh 4 #ad 4 #i4 #bgc qrs4 at #ais4 #b;d4 p] qu>t]8--,answ]1 #bfa 4 #ab4 #f4 #bfbg sacks at #gs4 #h;d4 p] sack8--,answ]1 #ajab 4 #i 4 #i4 #bca4 ,?r\gh\t ?is section x mu/ be obs]v$1 t ! rules c be appli$ to cases :]e ! quantities giv5 are express$ 9 common or decimal fractions1 in/ead ( ! measures 9 ! tables4 ,! foll[+ are examples3 ,:at is ! price ( #bgbcdgi cwt4 at #b4 #a4 #c p] cwt48 ,answ]1 #efbbhdi1 or #efb4 #e4 #h4 #fflbs4 at #as4 #d;d4 p] lb4 co/ #d4 #aa4 #e4 ,h[ _m p\nds1 %ill+s1 & p5ce1 w #bgicja acres let = if ea* acre lets = #cajgf8--;,answ]1 #hfgieeh1 or #hfg4 #ai4 #a4 ,:at does ( _/ ( #ag bu%4 co/ at ( ( #ag4 #ad p] bu%el8 ,answ]1 #bcadf1 or #b4 #f4 #c4 ,:at is ! co/ ( #ailbs4 #hoz4 #ab;dwt4 #h;gr4 at #d4 #d4 #f p] ounce8--,answ]1 #iii4 #ad4 #a 4 #bcb4 ,x is (t5 requir$ to f9d to h[ mu* a c]ta9 sum p] day w am\nt 9 a ye>4 ,?is may be %ortly done1 s9ce x happ5s t ! numb] ( days 9 a ye> is #bdj #abj #e2 s t a p5ny p] day is a p\nd1 half a p\nd1 & #e p5ce p] ye>4 ,h5ce ! foll[+ rule3 ,to f9d h[ mu* any sum p] day am\nts to 9 a ye>1 turn x into p5ce & fractions ( a p5ny2 to ?is add ! half ( itself1 & let ! p5ce be p\nds1 & ea* f>?+ five %ill+s2 !n add five times ! daily sum1 & ! total is ! ye>ly am\nt4 ,= example1 :at does #abs4 #c;d4 am\nt to 9 a ye>8 ,?is is #adg;d41 & its half is #gc;d41 :i* add$ to #adg;d4 gives #bba;d41 :i* turn$ into p\nds is #bba4 #ab4 #f4 ,also1 #abs4 #c;d4 #e is #c4 #a4 #f1 :i* add$ to ! =m] sum gives #bbd4 #ad4 #j = ! ye>ly am\nt4 ,9 ! same way ! ye>ly am\nt ( #bs4 #c;d4 is #da4 #af4 #e2 t ( #f;d4 is #aj4 #e4 #c2 & t ( #aa;d4 is #af4 #ad4 #g4 #bcc4 ,an inv]se rule may be =m$1 suffici5tly correct = e purpose1 9 ! foll[+ way3 ,if ! ye> consi/$ ( #cfj days1 or _/ ( #bdj1 ! subtraction ( "o-?ird f any sum p] ye> w\ld give ! proportion :i* belongs to #bdj days2 & e p\nd s obta9$ w\ld be "o p5ny p] day4 ,b z ! ye> is n #cfj1 b #cfe days1 if we divide ea* day's %>e into #cfe p>ts1 & take #e away1 ! :ole ( ! subtract$ sum1 or #cfj #e su* p>ts1 w give #cfj p>ts = ea* ( ! #e days :i* we neglect$ at fir/4 ,b #cfj su* p>ts are left beh9d = ea* ( ! #cfj fir/ days2 !re=e1 ?is additional process divides ! :ole annual am\nt equally among ! #cfe days4 ,n[1 #e p>ts out ( #cfe is "o out ( #gc1 or ! #gc;d "p ( ! fir/ result mu/ be subtract$ f x to produce ! true result4 ,unless ! daily sum be v l>ge1 ! #gb;d "p w d equally well1 :i*1 z #gb f>?+s are #ah p5ce1 is equival5t to subtract+ at ! rate ( "o f>?+ = #ah;d41 or d4 = #cs41 or #aj;d4 = #c4 ,! rule1 !n1 is z foll[s3 ,to f9d h[ mu* p] day w produce a giv5 sum p] ye>1 turn ! %ill+s1 c4 9 ! giv5 sum into decimals ( a p\nd 7#bba72 subtract "o-?ird2 consid] ! result z p5ce2 & dim9i% x by "o f>?+ = e eighte5 p5ce1 or t5 p5ce = e #c4 ,= example1 h[ mu* p] day w give #bbd4 #ad4 #j p] ye>8 ,?is is #bbdgjc1 & its ?ird is #gdija1 :i* subtract$ f #bbdgjc1 gives #adihjb1 :i*1 if !y be p5ce1 am\nts to #abs4 #ehjb;d41 9 :i* #as4 #f;d4 is conta9$ #h times4 ,subtract #h f>?+s1 or #b;d41 & we h #abs4 #chjb;d41 :i* diff]s f ! tru? only ab\t _/ ( a f>?+4 ,9 ! same way1 #ajj p] ye> is #es4 #e;d4 p] day4 #bcd4 ,! foll[+ connexion betwe5 ! measures ( l5g? & ! measures ( surface is ! f\ndation ( ! application ( ari?metic to geometry4 ,illu/ration ,suppose an oblong figure1 ,a1 ,b1 ,c1 ,d1 z "h drawn 7:i* is call$ a rectangle 9 geometry71 ) ! side ,a ,b #f in*es1 & ! side ,a ,c #d in*es4 ,divide ,a ,b & ,c ,d 7:i* are equal7 ea* into #f in*es by ! po9ts a1 b1 c1 l1 m1 c42 & ,a ,c & ,b ,d 7:i* are also equal7 into #d in*es by ! po9ts f1 g1 h1 x1 y1 & z4 ,jo9 a & l1 b & m1 c41 & f & x1 c4 ,!n1 ! figure ,a ,b ,c ,d is divid$ into a numb] ( squ>es2 = a squ>e is a rectangle ^: sides are equal1 & !re=e ,a a f ,e is squ>e1 s9ce ,a a is ( ! same l5g? z ,a f1 bo? be+ #a in*4 ,!re are also f\r r[s ( ^! squ>es1 ) six squ>es 9 ea* r[2 t is1 !re are #f #d1 or #bd squ>es altoge!r4 ,ea* ( ^! squ>es has its sides #a in* 9 l5g?1 & is :at 0 call$ 9 7#bae7 a squ>e in*4 ,by ! same reason+1 if "o side _h conta9$ #f y>ds1 & ! o!r #d y>ds1 ! surface w\ld h conta9$ #f #d squ>e y>ds2 & s on4 ,illu/ration #bce4 ,let u n[ suppose t ! sides ( ,a ,b ,c ,d1 in/ead ( be+ a :ole numb] ( in*es1 conta9 "s in*es & a fraction4 ,= example1 let ,a ,b be #c in*es1 or 7#aad7 _/ ( an in*1 & let ,a ,c conta9 #b in*es1 or _/ ( an in*4 ,draw ,a ,e twice z long z ,a ,b1 & ,a ,f f\r times z long z ,a ,c1 & complete ! rectangle ,a ,e ,f ,g4 ,! re/ ( ! figure ne$s no description4 ,!n1 s9ce ,a ,e is twice ,a ,b1 or twice _/ in*es1 x is #g in*es4 ,& s9ce ,a ,f is f\r times ,a ,c1 or f\r times _/ in*es1 x is #i in*es4 ,!re=e1 ! :ole rectangle ,a ,e ,f ,g conta9s1 by 7#bcd71 #g #i or #fc squ>e in*es4 ,b ! rectangle ,a ,e ,f ,g conta9s #h rectangles1 all ( ! same figure z ,a ,b ,c ,d2 & !re=e ,a ,b ,c ,d is "o-eigh? "p ( ,a ,e ,f ,g1 & conta9s _/ squ>e in*es4 ,b _/ is made by multiply+ _/ & _/ toge!r 7#aah74 ,f ?is & ! la/ article x appe>s1 t1 :e!r ! sides ( a rectangle be a :ole or a fractional numb] ( in*es1 ! numb] ( squ>e in*es 9 its surface is ! product ( ! numb]s ( in*es 9 its sides4 ,! squ>e itself is a rectangle ^: sides are all equal1 & !re=e ! numb] ( squ>e in*es :i* a squ>e conta9s is f\nd by multiply+ ! numb] ( in*es 9 its side by itself4 ,= example1 a squ>e ^: side is #ac in*es 9 l5g? conta9s #ac #ac or #afi squ>e in*es4 #bcf4 ,,ex]cises4 ,:at is ! cont5t1 9 squ>e feet & in*es1 ( a room ^: sides are #db ft4 #e in*4 & #ca ft4 #i in*48 & suppos+ ! piece f :i* its c>pet is tak5 to be ?ree qu>t]s ( a y>d 9 bread?1 :at l5g? ( x mu/ be cut (f8--,answ]1 ,! cont5t is #acdf squ>e feet #aje squ>e in*es1 & ! l5g? ( c>pet requir$ is #eih feet #f_/ in*es4 ,! sides ( a rectangul> field are #bec y>ds & a qu>t] ( a mile2 h[ _m acres does x conta98--,answ]1 #bc4 ,:at is ! diff]5ce betwe5 #ah squ>e miles1 & a squ>e ( #ah miles long1 or #ah miles squ>e8--,answ]1 #cjf squ>e miles4 #bcg4 ,x is by ?is rule t ! measure 9 7#bae7 is d$uc$ f t 9 7#bad72 = x is evid5t t twelve in*es be+ a foot1 ! squ>e foot is #ab #ab or #add squ>e in*es1 & s on4 ,9 a simil> way x may be %ewn t ! cont5t 9 cubic in*es ( a cube1 or p>allelepip$1#dh may be f\nd by multiply+ toge!r ! numb] ( in*es 9 ^? ?ree sides :i* meet 9 a po9t4 ,?us1 a cube ( #f in*es conta9s #f #f #f1 or #baf cubic in*es2 a *e/ ^: sides are #f1 #h1 & #e feet1 conta9s #f #h #e1 or #bdj cubic feet4 ,by ?is rule ! measure 9 7#baf7 0 d$uc$ f t 9 7#bad74 #dh ,a p>allelepip$1 or m prop]ly1 a rectangul> p>allelepip$1 is a figure ( ! =m ( a brick2 its sides1 h[ev]1 may be ( any l5g?2 ?us1 ! figure ( a plank has ! same "n4 ,a cube is a p>allelepip$ ) equal sides1 su* z is a die4 ,,section ,,ii4 ,,rule ,,( ,,?ree4 #bch4 ,suppose x requir$ to f9d :at #aef y>ds w co/1 if #bb y>ds co/ #ags4 #d;d4 ,?is quantity1 r$uc$ to p5ce1 is #bjh;d42 & if #bb y>ds co/ #bjh;d41 ea* y>d co/s _/d4 ,b #aef y>ds co/ #aef times ! price ( "o y>d1 & !re=e co/ #bjh #bjh #aef ---- #aef p5ce1 or --------- p5ce 7#aag74 #bb #bb ,aga91 if #be ,fr5* francs be #bj %ill+s /]l+1 h[ _m francs are 9 #bj4 #ae8 ,s9ce #be francs are #bj %ill+s1 twice ! numb] ( francs mu/ be twice ! numb] ( %ill+s2 t is1 #ea francs are #dj %ill+s1 & "o %ill+ is ! =tie? "p ( #ea francs1 or _/ francs4 ,b #bj #aes4 conta9 #dae %ill+s 7#bai72 & s9ce #a %ill+ is _/ francs1 #dae %ill+s is #ea #dae _/ #dae francs1 or 7#aag7 -------- francs4 #dj #bci4 ,su* que/ions z ! la/ two belong to ! mo/ ext5sive rule 9 ,comm]cial ,ari?metic1 :i* is call$ ! ,,rule ,,( ,,?ree1 because 9 x ?ree quantities are giv51 & a f\r? is requir$ to be f\nd4 ,f bo? ! prec$+ examples ! foll[+ rule may be d$uc$1 :i* ! same reason+ w %ew to apply to all simil> cases4 ,x mu/ be obs]v$1 t 9 ^! que/ions !re are two quantities :i* are ( ! same sort1 & a ?ird ( ano!r sort1 ( :i* la/ ! answ] mu/ be4 ,?us1 9 ! fir/ "q !re are #bb & #aef y>ds & #bjh p5ce1 & ! ?+ requir$ to be f\nd is a numb] ( p5ce4 ,9 ! second "q !re are #bj & #dae %ill+s & #be francs1 & :at is to be f\nd is a numb] ( francs4 ,write ! ?ree quantities 9 a l9e1 putt+ t "o la/ :i* is ! only "o ( its k9d1 & t "o fir/ :i* is connect$ ) ! la/ 9 ! "q4#di ,put ! ?ird quantity 9 ! middle4 ,9 ! fir/ "q ! quantities w be plac$ ?us3 #bb yds4 #aef yds4 #ags4 #d;d4 ,9 ! second "q !y w be plac$ ?us3 #bjs4 #bj #aes4 #be francs4 #di ,?is g5]ally comes 9 ! same memb] ( ! s5t5ce4 ,9 "s cases ! +5uity ( ! /ud5t mu/ be employ$ 9 detect+ x4 ,! reason+ ( 7#bch7 is ! be/ guide4 ,! foll[+ may be v (t5 appli$4 ,if x be evid5t t ! answ] mu/ be less ?an ! giv5 quantity ( its k9d1 multiply t giv5 quantity by ! less ( ! o!r two2 if great]1 by ! great]4 ,?us1 9 ! fir/ "q1 #aef y>ds mu/ co/ m ?an #bb2 multiply1 !re=e1 by #aef4 ,r$uce ! fir/ & second quantities1 if necess>y1 to quantities ( ! same d5om9ation4 ,?us1 9 ! second "q1 #bj #aes4 mu/ be r$uc$ to %ill+s 7#bai74 ,! ?ird quantity may also be r$uc$ to any o!r d5om9ation1 if conv5i5t2 or ! fir/ & ?ird may be multipli$ by any quantity we please1 z 0 done 9 ! second "q2 &1 on look+ at ! answ] 9 7#bch71 & at 7#ajh71 x w be se5 t no *ange is made by t multiplication4 ,multiply ! second & ?ird quantities toge!r1 & divide by ! fir/4 ,! result is a quantity ( ! same sort z ! ?ird 9 ! l9e1 & is ! answ] requir$4 ,?us1 to ! fir/ "q ! answ] is 7#bch7 #bjh #aef #ags4 #d;d4 #aef ----------p5ce1 or1 :i* is ! same ?+1 -------------------4 #bb #bb #bdj4 ,! :ole process 9 ! fir/ "q is z foll[s3#ej yds4 yds4 s4 d4 #bb 3 #aef #ag 4 #d #ab --- #bjh p5ce4 #aef ---- #abdh #ajdj #bjh ----- #bb7cbddh7adgd;d4 & _/1 or _/ ( a f>?+1 #bb or 7#bai7 #f 4 #b 4 #aj-_/4 --- #ajd #hh ---- #afd #aed ---- #ajh #hh -- #bj 7#bbh7 #d -- #hj #ff -- #ad #ej ,x is usual to place po9ts1 9 ! mann] "h %ewn1 betwe5 ! quantities4 ,^? :o h read ,section ,,viii4 w see t ! ,rule ( ,?ree is no m ?an ! process = f9d+ ! f\r? t]m ( a proportion f ! o!r ?ree4 ,! "q might h be5 solv$ )\t r$uc+ #ags4 #d;d4 to p5ce1 ?us3 yds4 yds4 s4 d4 #bb 3 #aef #ag 4 #d #aef 7#bbg7 ---------- #bb7ace 4 #d 4 #j7f 4 #b 4 #aj-_/ 7#bbh7 #acb --- #c #bj #d #fd #dd -- #bj #ab #bdj #bbj --- #bj #d #hj #ff -- #ad ,! /ud5t mu/ le>n by practice :i* is ! mo/ conv5i5t me?od = any p>ticul> case1 z no rule c be giv54 #bda4 ,x may happ5 t ! ?ree giv5 quantities are all ( "o d5om9ation2 nev]!less x w be f\nd t two ( !m are ( "o1 & ! ?ird ( ano!r sort4 ,= example3 ,:at mu/ an income ( #djj pay t[>ds an income-tax ( #ds4 #f;d4 9 ! p\nd8 ,"h ! ?ree giv5 quantities are1 #djj1 #ds4 #f;d41 & #a1 :i* are all ( ! same species1 viz4 money4 ,nev]!less1 ! fir/ & ?ird are income2 ! second is a tax1 & ! answ] is also a tax2 & !re=e1 by 7#aeb71 ! quantities mu/ be plac$ ?us3 #a 3 #djj #ds4 #f;d4 #bdb4 ,! foll[+ ex]cises ei!r dep5d directly ^u ?is rule1 or c be %ewn to d s by a little consid]ation4 ,!re are _m que/ions ( ! sort1 :i* w require "s ex]cise ( +5uity be=e ! me?od ( apply+ ! rule c be f\nd4 ,,ex]cises4 ,if #ae cwt4 #b qrs4 co/ #aih4 #ae4 #d1 :at does #a qr4 #bb lbs4 co/8 ,answ]1 #e 4 #ad 4 #e _/4 ,if a horse g #ad m4 #c fur4 #bg yds4 9 #c #bf #ab1 h[ long w he be 9 go+ #bc miles8 ,answ]1 #e #bi #cd7_/74 ,two p]sons1 ,a & ,b1 are bankrupts1 & owe exactly ! same sum2 ,a c pay #aes4 #d;d4 9 ! p\nd1 & ,b only #gs4 7#f7;d4 ,at ! same "t ,a has 9 8 possession #acjd4 #ag m ?an ,b2 :at d ! debts ( ea* am\nt to8 ,answ]1 #ccdj 4 #h 4 #c _/4 ,= e 7#ab7 acres :i* "o c\ntry conta9s1 a second conta9s 7#ef74 ,! second c\ntry conta9s #ag1cjj squ>e miles4 ,h[ mu* does ! fir/ conta98 ,aga91 = e #c p 9 ! fir/1 !re are #e 9 ! second2 & !re are 9 ! fir/ #bg p on e #bj acres4 ,h[ _m are !re 9 ea* c\ntry8--,answ]1 ,! numb] ( squ>e miles 9 ! fir/ is #chdd_/1 & its population #c1cba1fjj2 & ! population ( ! second is #e1ecf1jjj4 ,if 7#db7 yds4 ( clo?1 #ah 94 wide1 co/ #ei4 #ad4 #b1 h[ mu* w 7#aah7 yds4 co/1 if ! wid? be #a yd48 ,answ]1 #ccb4 #e4 7#b_/74 ,if #i4 #c4 #f la/ six weeks1 h[ long w #ajj la/8 ,answ]1 7#fe_/7 weeks4 ,h[ mu* sug>1 wor? 7#i;d74 a p\nd1 mu/ be giv5 = #b cwt4 ( tea1 wor? #aj;d4 an ounce8 ,answ]1 #cb cwt4 #c qrs4 #g lbs4 _/4 #bdc4 ,suppose ! foll[+ "q ask$3 ,h[ long w x take #ae m5 to d t :i* #de m5 c f9i% 9 #aj days8 ,x is evid5t t "o man w\ld take #de #aj1 or #dej days1 to d ! same ?+1 & t #ae m5 w\ld d x 9 "o-fifte5? "p ( ! "t :i* x employs "o man1 t is1 9 7#dej #ae7 or #cj days4 ,by ?is & simil> reason+ ! foll[+ que/ions c be solv$4 ,,ex]cises4 ,if #ae ox5 eat an acre ( grass 9 #ab days1 h[ long w x take #bf ox5 to eat #ad acres8 ,answ]1 7#if_/7 days4 ,if #bb masons build a wall #e feet high 9 #f days1 h[ long w x take #dc masons to build #aj feet8 ,answ]1 7#f_/7 days4 #bdd4 ,! que/ions 9 ! prec$+ article =m "p ( a m g5]al class ( que/ions1 ^: solution is call$ ! ,,d\ble ,,rule ,,( ,,?ree1 b :i* might1 ) m correctness1 be call$ ! ,rule ( ,five1 s9ce five quantities are giv51 & a six? is to be f\nd4 ,! foll[+ is an example3 ,if #e m5 c make #cj y>ds ( clo? 9 #c days1 h[ long w x take #d m5 to make #fh y>ds8 ,! fir/ ?+ to be done is to f9d out1 f ! fir/ "p ( ! "q1 ! "t x w take "o man to make "o y>d4 ,n[1 s9ce "o man1 9 #c days1 w d ! fif? "p ( :at #e m5 c d1 he w 9 #c days make _/1 or #f y>ds4 ,he w1 !re=e1 make "o y>d 9 _/#f or 7#c #e7_/cj ( a day4 ,f ?is we are to f9d h[ long x w take #d m5 to make #fh y>ds4 ,s9ce "o man makes a y>d 9 #c #e #c #e ----- ( a day1 he w make #fh y>ds 9 ----- #fh days1 #cj #cj #c #e #fh or 7#aaf7 9 ---------- days2 & #d m5 w d ?is 9 "o-f\r? #cj #c #e #fh ( ! "t1 t is 7#abc71 9 ---------- days1 or 9 #h days4 #cj #d ,aga91 suppose ! "q to be3 ,if #e m5 c make #cj y>ds 9 #c days1 h[ mu* c #f m5 d 9 #ab days8 ,"h we mu/ fir/ f9d ! quantity "o man c d 9 "o day1 :i* appe>s1 on reason+ simil> to t 9 ! la/ example1 to be #cj_/7c #e7 y>ds4 ,h5ce1 #f m51 9 "o day1 w make #f #cj #ab #f #cj ------ y>ds1 & 9 #ab days w make ----------- or #add y>ds4 #e #c #e #c ,f ^! examples ! foll[+ rule may be drawn4 ,write ! giv5 quantities 9 two l9es1 keep+ quantities ( ! same sort "u "o ano!r1 & ^? :i* are connect$ ) ea* o!r1 9 ! same l9e4 ,9 ! two examples above giv51 ! quantities mu/ be writt5 ?us3 ,illu/ration ,,second ,,example4 ,illu/ration ,draw a curve ?r\gh ! middle ( ea* l9e1 & ! extremities ( ! o!r4 ,!re w be ?ree quantities on "o curve & two on ! o!r4 ,divide ! product ( ! ?ree by ! product ( ! two1 & ! quoti5t is ! answ] to ! "q4 ,if necess>y1 ! quantities 9 ea* l9e mu/ be r$uc$ to m simple d5om9ations 7#bai71 z 0 done 9 ! common ,rule ( ,?ree 7#bch74 ,,ex]cises4 ,if #f horses c1 9 #b days1 pl\gh #ag acres1 h[ _m acres w #ic horses pl\gh 9 #d days8 ,answ]1 #eib4 ,if #bj m51 9 #c days1 c dig #g rectangul> fields1 ! sides ( ea* ( :i* are #dj & #ej y>ds1 h[ long w #cg m5 be 9 digg+ #ec fields1 ! sides ( ea* ( :i* are #ij & #abe y>ds8 #bdea ,answ]1 #ge----- days4 #bjgbj ,if ! c>riage ( #fj cwt4 ?r\gh #bj miles co/ #ad #ajs41 :at weight ought to be c>ri$ #cj miles = #e4 #h4 #i8 ,answ]1 #ae cwt4 ,if #ajj ga9 #e 9 a ye>1 h[ mu* w #hej ga9 9 #c ye>s & #h mon?s8 ,answ]1 #aee4 #af4 #h4 ,,section ,,iii4 ,,int]e/1 ,,etc4 #bde4 ,9 ! que/ions conta9$ 9 ?is ,section1 almo/ ! only process :i* w be employ$ is ! tak+ a fractional "p ( a sum ( money1 :i* has be5 done be=e 9 sev]al cases4 ,suppose x requir$ to take #g p>ts out ( #dj f #af1 t is1 to divide #af into #dj equal p>ts1 & take #g ( !m4 ,ea* ( ^! p>ts is #af #af #af #g ----1 & #g ( !m make ---- #g1 or ------ p\nds 7#aaf74 #dj #dj #dj ,! process may be writt5 z bel[3 #af #g ----- #dj7aab7b 4 #afs4 #hj -- #cb #bj --- #fdj #dj --- #bdj #bdj --- #j ,suppose x requir$ to take #ac p>ts out ( a hundr$ f #ef4 #ac4 #g4 #ef 4 #ac 4 #g #ac ---------------- #ajj7 #gcf 4 #ag 4 #a 7 #g 4 #g 4 #d _/ #gjj --- #cf #bj #ag #gcg #gjj --- #cg #ab #a #dde #djj --- #de #d #b #ahb #ajj --- #hb ,let x be requir$ to take #b p>ts out ( a hundr$ f #c #abs4 ,! result1 by ! same rule is #c #abs4 #b #e -------------------1 or #abc #c #abs4 ---2 #ajj #bjj s t tak+ #b out ( a hundr$ is ! same z tak+ #e p>ts out ( #bjj4 ,,ex]cises4 ,take #g p>ts out ( #ec f #a #ajs4 #abi ,answ]1 #ds4 #a---;d4 #aei ,take #e p>ts out ( #ajj f #ajg #acs4 #d;d4 ,answ]1 #e4 #g4 #h & _/ ( a f>?+4 #ef #cs4 #b;d4 is equally divid$ among #cb p]sons4 ,h[ mu* does ! %>e ( #bc ( !m exce$ t ( ! re/8 ,answ]1 #bd4 #aa4 #d 4 #bdf4 ,x is usual1 9 m]cantile bus9ess1 to m5tion ! fraction :i* "o sum is ( ano!r1 by say+ h[ _m p>ts out ( a hundr$ mu/ be tak5 f ! second 9 ord] to make ! fir/4 ,?us1 in/ead ( say+ t #af #abs4 is ! half ( #cc #ds41 x is said t ! fir/ is #ej p] c5t ( ! second4 ,?us1 #e is #b p] c5t ( #bjj2 because1 if #bjj be divid$ into #ajj p>ts1 #b ( ^? p>ts are #e4 ,also1 #ac is #aej p] c5t ( #h4 #ac4 #d1 s9ce ! fir/ is ! second & half ! second4 ,suppose x ask$1 ,h[ mu* p] c5t is #bc p>ts out ( #ef ( any sum8 ,! "q am\nts to ?is3 ,if he :o has #ef gets #ajj = !m1 h[ mu* w he :o has #bc receive8 ,?is1 by #bch1 is #bc _/ or _/ or #da_/4 ,h5ce1 #bc out ( #ef is #da_/ p] c5t4 ,simil>ly #af p>ts out ( #ah is #af _/1 or #hh_/ p] c5t1 & #b p>ts out ( #e is #b _/1 or #dj p] c5t4 ,f :i* ! me?od ( r$uc+ o!r fractions to ! rate p] c5t is evid5t4 ,suppose x ask$1 ,h[ mu* p] c5t is #f4 #ab4 #b ( #ab4 #c8 ,s9ce ! fir/ conta9s #aehf;d41 & ! second #biaf;d41 ! fir/ is #aehf out ( #biaf p>ts ( ! second2 t is1 by ! la/ rule1 x is _/1 or #ed_/1 or #ed4 #g4 #i p] c5t1 v ne>ly4 ,! m exp$iti\s way ( do+ ?is is to r$uce ! %ill+s1 c4 to decimals ( a p\nd4 ,?ree decimal places w give ! rate p] c5t to ! ne>e/ %ill+1 :i* is ne> en\gh = all practical purposes4 ,= in/ance1 9 ! la/ example1 :i* is to f9d h[ mu* #ffjh is ( #abae1 #ffjh #ajj is #ffjh1 :i* divid$ by #abae gives #edch1 or #ed4 #g4 ,great] correctness may be _h1 if necess>y1 z 9 ! ,app5dix4 ,,ex]cises4 ,h[ mu* p] c5t is #aih out ( #bcc p>ts8--,ans4 #he4 #a4 #h4 ,goods :i* are b\ght = #aic4 #ab1 are sold = #baf4 #ac4 #d2 h[ mu* p] c5t has be5 ga9$ by !m8 ,answ]1 ,a little less ?an #aa4 #ah4 #f4 ,a sells goods = ,b to ! am\nt ( #bcj4 #ab1 & is all[$ a commission#ea ( #c p] c5t2 :at does t am\nt to8 ,answ]1 #f 4 #ah4 #d _/4 #ea ,commission is :at is all[$ by "o m]*ant to ano!r = buy+ or sell+ goods = him1 & is usually a p]-c5tage on ! :ole sum employ$4 ,brok]age is an all[ance simil> to commission1 "u a diff]5t "n1 pr9cipally us$ 9 ! buy+ & sell+ ( /ock 9 ! funds4 ,insurance is a p]-c5tage paid to ^? :o engage to make good to ! pay]s any loss !y may su/a9 by accid5ts f fire1 or /orms1 accord+ to ! agreem5t1 up to a c]ta9 am\nt :i* is nam$1 & is a p]-c5tage ^u ?is am\nt4 ,t>e1 tret1 & cl(f1 are all[ances made 9 sell+ goods by :olesale1 = ! weight ( ! boxes or b>rels :i* conta9 !m1 wa/e1 c42 & are usually ei!r ! price ( a c]ta9 numb] ( p\nds ( ! goods = ea* box or b>rel1 or a c]ta9 all[ance on ea* cwt4 ,a /ockbrok] buys #agjj /ock1 brok]age be+ at p] c5t2 :at does he receive8--,answ]1 #b4 #b4 #f4 ,a %ip ^: value is #ae1dbc is insur$ at #ai p] c5t2 :at does ! insurance am\nt to8--,answ]1 #cjcc4 #c4 #i _/4 #bdg4 ,9 reckon+ h[ mu* a bankrupt is able to pay 8 cr$itors1 z also to h[ mu* a tax or rate am\nts1 x is usual to f9d h[ _m %ill+s 9 ! p\nd is paid4 ,?us1 if a p]son :o owes #ajj c only pay #ej1 he is said to pay #ajs4 9 ! p\nd4 ,! rule is easily d]iv$ f ! same reason+ z 9 #bdf4 ,= example1 #ej out ( #hb is #ej #ejbj ---- out ( #a1 or ----- %ill+s1 #hb #hb or #abs4 #b _/ 9 ! p\nd4 #bdh4 ,,int]e/ is money paid = ! use ( o!r money1 & is always a p]-c5tage ^u ! sum l5t4 ,x may be paid ei!r ye>ly1 half-ye>ly1 or qu>t]ly2 b :5 x is said t #ajj is l5t at #d p] c5t1 x mu/ be und]/ood to mean #d p] c5t p] annum2 t is1 t #d p\nds are paid e ye> = ! use ( #ajj4 ,! sum l5t is call$ ! pr9cipal1 & ! int]e/ ^u x is ( two k9ds4 ,if ! borr[] pay ! int]e/ z soon z1 f ! agreem5t1 x becomes due1 x is evid5t t he has to pay ! same sum e ye>2 & t ! :ole ( ! int]e/ :i* he has to pay 9 any numb] ( ye>s is "o ye>'s int]e/ multipli$ by ! numb] ( ye>s4 ,b if he d n pay ! int]e/ at once1 b keeps x 9 8 h&s until he returns ! pr9cipal1 he w !n h m ( 8 cr$itor's money 9 8 h&s e ye>1 & if x 7 s agre$ w h to pay int]e/ ^u ea* ye>'s int]e/ = ! "t dur+ :i* he keeps x aft] x becomes due4 ,9 ! fir/ case1 ! int]e/ is call$ simple1 & 9 ! second comp\nd4 ,! int]e/ & pr9cipal toge!r are call$ ! am\nt4 #bdi4 ,:at is ! simple int]e/ ( #ajdi4 #af4 #f = #f ye>s & "o-?ird1 at #d p] c5t8 ,?is int]e/ mu/ be #f times ! int]e/ ( ! same sum = "o ye>1 :i* 7#bde7 is f\nd by multiply+ ! sum by #d1 & divid+ by #ajj4 ,! process is z foll[s3 7#bcj7 7a7 #ajdi 4 #af 4 #f -------------- a #d #daii 4 #f 4 #j a #ebd 4 #ah 4 #c -------------- 7#hb7 #ajj7 #dg1bd 4 #d 4 #c7dg 4 #d 4 #aj_/ #bj ---- 7#bbh7 #d1hdeb #ab ------ #aj1aaec 7b7 #dg 4 #d 4 #aj_/ ,int4 = "o yr4 ------------------ b #f #bhc 4 #i 4 #j_/ b #ae 4 #ad 4 #aa_/ --------------------- #bii 4 #d 4 #j_/ ,int4 = #f yrs4 #eb ,"h ! #ds4 f ! divid5d is tak5 94 #ec ,"h ! #c;d4 f ! divid5d is tak5 94 ,,ex]cises4 ,:at is ! int]e/ ( #aje4 #f4 #b = #ai ye>s & #g weeks at #c p] c5t8 ,answ]1 #fj4 #i1 v ne>ly4 ,:at is ! diff]5ce betwe5 ! int]e/ ( #ej4 #ai = #g ye>s at #c p] c5t1 & = #h ye>s at #b p] c5t8 ,answ]1 #ajs4 7#b7;d4 ,:at is ! int]e/ ( #aeg4 #ag4 #f = "o ye> at #e p] c5t8 ,answ]1 #g4 #ag4 #aj4 ,%ew t ! int]e/ ( any sum = #i ye>s at #d p] c5t is ! same z t ( ! same sum = #d ye>s at #i p] c5t4 #bej4 ,9 ord] to f9d ! int]e/ ( any sum at comp\nd int]e/1 x is necess>y to f9d ! am\nt ( ! pr9cipal & int]e/ at ! end ( e ye>2 because 9 ?is case 7#bdh7 x is ! am\nt ( bo? pr9cipal & int]e/ at ! end ( ! fir/ ye>1 ^u :i* int]e/ accumulates dur+ ! second ye>4 ,suppose1 = example1 x is requir$ to f9d ! int]e/1 = #c ye>s1 on #ajj1 at #e p] c5t1 comp\nd int]e/4 ,! foll[+ is ! process3 #ajj ,fir/ pr9cipal4 #e ,fir/ ye>'s int]e/4 --- #aje ,am\nt at ! end ( ! fir/ ye>4 7#bdi7 #e 4 #e ,int]e/ = ! second ye> on #aje4 -------- #aaj 4 #e ,am\nt at ! end ( two ye>s4 #e 4 #aj 4 #c ,int]e/ due = ! ?ird ye>4 ------------ #aae 4 #ae 4 #c ,am\nt at ! end ( ?ree ye>s4 #ajj 4 #j 4 #j ,fir/ pr9cipal4 ------------ #ae 4 #ae 4 #c ,int]e/ ga9$ 9 ! ?ree ye>s4 ,:5 ! numb] ( ye>s is great1 & ! sum consid]able1 ?is process is v tr\blesome2 on :i* acc\nt tables#ed are con/ruct$ to %ew ! am\nt ( "o p\nd1 = diff]5t numb]s ( ye>s1 at diff]5t rates ( int]e/4 ,to make use ( ^! tables 9 ! pres5t example1 look into ! column head$ 8#e p] c5t20 & opposite to ! numb] #c1 9 ! column head$ 8,numb] ( ye>s10 is f\nd #aaegfbe2 mean+ t #a w become #aaegfbe 9 #c ye>s4 ,n[1 #ajj mu/ become #ajj times z great2 & #aaegfbe #ajj is #aaegfbe 7#ada72 b 7#bba7 #gfbe is #aes4 #c;d42 !re=e ! :ole am\nt ( #ajj is #aae4 #ae4 #c1 z be=e4 #ed ,suffici5t tables = all common purposes are conta9$ 9 ! article on ,int]e/ 9 ! ,p5ny ,cyclopdia2 & ample ones 9 ! ,treatise on ,annuities & ,rev]sions1 9 ! ,libr>y ( ,useful ,k4 #bea4 ,suppose t a sum ( money has la9 at simple int]e/ #d ye>s1 at #e p] c5t1 & has1 ) its int]e/1 am\nt$ to #cej2 x is requir$ to f9d :at ! sum 0 at fir/4 ,:atev] ! sum 01 if we suppose x divid$ into #ajj p>ts1 #e ( ^? p>ts 7 add$ e ye> = #d ye>s1 z int]e/2 t is1 #bj ( ^? p>ts h be5 add$ to ! fir/ sum to make #cej4 ,if1 !re=e1 #cej be divid$ into #abj p>ts1 #ajj ( ^? p>ts are ! pr9cipal :i* we want to f9d1 & #bj p>ts are int]e/ ^u x2 t is1 ! pr9cipal is 7#cej #ajj7_/aej1 or #bia4 #ac4 #d4 #beb4 ,suppose t ,a 0 engag$ to pay ,b #cej at ! end ( f\r ye>s f ?is "t1 & t x is agre$ betwe5 !m t ! debt %all be paid imm$iately2 suppose1 also1 t money c be employ$ at #e p] c5t1 simple int]e/2 x is pla9 t ,a ought n to pay ! :ole sum1 #cej1 because1 if he did1 he w\ld lose #d ye>s' int]e/ ( ! money1 & ,b w\ld ga9 x4 ,x is fair1 !re=e1 t he %\ld only pay to ,b z mu* z w1 ) int]e/1 am\nt 9 f\r ye>s to #cej1 t is 7#bea71 #bia4 #ac4 #d4 ,!re=e1 #eh4 #f4 #h mu/ be /ruck (f ! debt 9 consid]ation ( its be+ paid be=e ! "t4 ,?is is call$ ,,disc\nt2#ee & #bia4 #ac4 #d is call$ ! pres5t value ( #cej due f\r ye>s h5ce1 disc\nt be+ at #e p] c5t4 ,! rule = f9d+ ! pres5t value ( a sum ( money 7#bea7 is3 ,multiply ! sum by #ajj1 & divide ! product by #ajj increas$ by ! product ( ! rate p] c5t & numb] ( ye>s4 ,if ! "t t ! debt has yet to run be express$ 9 ye>s & mon?s1 or mon?s only1 ! mon?s mu/ be r$uc$ to ! equival5t fraction ( a ye>4 #ee ,?is rule is obsolete 9 bus9ess4 ,:5 a bill1 = in/ance1 ( #ajj hav+ a ye> to run1 is disc\nt$ 7z p n[ say7 at #e p] c5t1 ?is means t #e p] c5t ( #ajj1 or #e1 is /ruck (f4 ,,ex]cises4 ,:at is ! disc\nt on a bill ( #ach4 #ad4 #d1 due #b ye>s h5ce1 disc\nt be+ at #d p] c5t8 ,answ]1 #aa4 #i4 #a4 ,:at is ! pres5t value ( #ajca4 #ag1 due #f mon?s h5ce1 int]e/ be+ at #c p] c5t8 ,answ]1 #ajaf4 #ab4 #bec4 ,if we multiply by a b1 or by a-b1 :5 we %\ld multiply by a1 ! result is wrong by ! fraction b b --- b1 or ---------1 a a - b ( itself3 be+ too great 9 ! fir/ case1 & too small 9 ! second4 ,aga91 if we divide by a b1 :]e we %\ld h divid$ by a1 ! result is too small by ! fraction b_/a ( itself2 :ile1 if we divide by a-b in/ead ( a1 ! result is too great by ! same fraction ( itself4 ,?us1 if we divide by #bj in/ead ( #ag1 ! result is _/ ( itself too small2 & if we divide by #cfj in/ead ( #cfe1 ! result is too great by _/1 or _/ ( itself4 ,if1 !n1 we wi% to f9d ! int]e/ ( a sum ( money = a portion ( a ye>1 & h n ! assi/ance ( tables1 x w be f\nd conv5i5t to suppose ! ye> to conta9 only #cfj days1 9 :i* case its #gc;d "p 7! #gb;d "p w g5]ally d7 mu/ be subtract$ f ! result1 to make ! alt]ation ( #cfj into #cfe4 ,! numb] #cfj has s l>ge a numb] ( divisors1 t ! rule ( ,practice 7#bcj7 may always be readily appli$4 ,?us1 x is requir$ to f9d ! portion :i* belongs to #bgd days1 ! ye>ly int]e/ be+ #ah4 #i4 #aj1 or #ahdia4 #bgd #ahdia ------ #ahj is ( #cfj #ibdf --- #id #ij is ( #ahj #dfbc -- #d is _/ ( #cfj #bje ------ #i7adjgd ------ #h7aefd ----- #aif #achgh #ac 4 #ag 4 #g ,answ]4 ,b if ! ne>e/ f>?+ be want$1 ! be/ way is to take #b-t5?s ( ! numb] ( days z a multipli]1 & #gc z a divisor2 s9ce m #cfe is #bm #gcj1 or 7_/7m #gc4 ,?us1 9 ! prec$+ in/ance1 we multiply by #edh & divide by #gc2 & #edh #ahdia #ajaccjfh1 :i* divid$ by #gc gives #achha1 v ne>ly agree+ ) ! =m]1 & giv+ #ac4 #ag4 #g1 :i* is c]ta9ly )9 a f>?+ ( ! tru?4 #bed4 ,suppose x requir$ to divide #ajj among ?ree p]sons 9 su* a way t _! %>es may be z #f1 #e1 & #i2 t is1 s t = e #f :i* ! fir/ has1 ! second may h #e1 & ! ?ird #i4 ,x is pla9 t if we divide ! #ajj into #f #e #i1 or #bj p>ts1 ! fir/ mu/ h #f ( ^? p>ts1 ! second #e1 & ! ?ird #i4 ,!re=e 7#bde7 _! %>es are respectively1 #ajj #f #ajj #e #ajj #i -------1 ------- & -------1 or #cj1 #be1 & #de4 #bj #bj #bj ,,ex]cises4 ,divide #cid4 #ab among f\r p]sons1 s t _! %>es may be z #a1 #f1 #g1 & #ah4--;,answ]1 #ab4 #f4 #g2 #gc4 #ai4 #i2 #hf4 #f4 #d2 #bba4 #ai4 #c4 ,divide #bj among #f p]sons1 s t ! %>e ( ea* may be z mu* z ^? ( all :o come be=e put toge!r4--,answ]1 ,! fir/ two h #abs4 #f;d42 ! ?ird #a4 #e2 ! f\r? #b4 #aj2 ! fif? #e2 & ! six? #aj4 #bee4 ,:5 two or m p]sons employ _! money toge!r1 & ga9 or lose a c]ta9 sum1 x is evid5tly n fair t ! ga9 or loss %\ld be equally divid$ among !m all1 unless ea* contribut$ ! same sum4 ,suppose1 = example1 ,a contributes twice z mu* z ,b1 & !y ga9 #ae1 ,a ought to ga9 twice z mu* z ,b2 t is1 if ! :ole ga9 be divid$ into #c p>ts1 ,a ought to h two ( !m & ,b "o1 or ,a %\ld ga9 #aj & ,b #e4 ,suppose t ,a1 ,b1 & ,c engage 9 an adv5ture1 9 :i* ,a emb>ks #bej1 ,b #acj1 & ,c #de4 ,!y ga9 #ajjj4 ,h[ mu* ( x ought ea* to h8 ,ea* "o ought to ga9 z mu* = #a z ! o!rs4 ,n[1 s9ce !re are #bej #acj #de1 or #dbe p\nds emb>k$1 :i* ga9 #ajjj1 = ea* p\nd !re is a ga9 ( _/4 ,!re=e ,a %\ld ga9 #ajjj _/ p\nds1 ,b %\ld ga9 #ajjj _/ p\nds1 & ,c #ajjj _/ p\nds4 ,on ^! pr9ciples1 by ! process 9 7#bde71 ! foll[+ que/ions may be answ]$4 ,a %ip is to be insur$1 9 :i* ,a has v5tur$ #aibh1 & ,b #difc4 ,! exp5se ( insurance is #dgd4 #aj4 #b4 ,h[ mu* ought ea* to pay ( x8 ,answ]1 ,a mu/ pay #acb4 #ae4 7#b74 ,a loss ( #adi is to be made good by ?ree p]sons1 ,a1 ,b1 & ,c4 ,_h !re be5 a ga91 ,a w\ld h ga9$ #d times z mu* z ,b1 & ,c z mu* z ,a & ,b toge!r4 ,h[ mu* ( ! loss mu/ ea* be>8 ,answ]1 ,a pays #ei4 #ab1 ,b #ad4 #ah1 & ,c #gd4 #aj4 #bef4 ,x may happ5 t sev]al individuals employ sev]al sums ( money toge!r = diff]5t times4 ,9 su* a case1 unless !re be a special agreem5t to ! contr>y1 x is "r t ! m "t a sum is employ$1 ! m pr(it %\ld be made ^u x4 ,if1 = example1 ,a & ,b employ ! same sum = ! same purpose1 b ,a's money is employ$ twice z long z ,b's1 ,a ought to ga9 twice z mu* z ,b4 ,! pr9ciple is1 t "o p\nd employ$ = "o mon?1 or "o ye>1 ought to give ! same return to ea*4 ,suppose1 = example1 t ,a employs #c = #f mon?s1 ,b #d = #g mon?s1 & ,c #ab = #b mon?s1 & ! ga9 is #ajj2 h[ mu* ought ea* to h ( x8 ,n[1 s9ce ,a employs #c = six mon?s1 he mu/ ga9 #f times z mu* z if he employ$ x "o mon? only2 t is1 z mu* z if he employ$ #f #c1 or #ah1 = "o mon?2 also1 ,b ga9s z mu* z if he _h employ$ #d #g = "o mon?2 & ,c z if he _h employ$ #ab #b = "o mon?4 ,if1 !n1 we divide #ajj into #f #c #d #g #ab #b1 or #gj p>ts1 ,a mu/ h #f #c1 or #ah1 ,b mu/ h #d #g1 or #bh1 & ,c #ab #b1 or #bd ( ^? p>ts4 ,! %>es ( ! ?ree are1 !re=e1 #f #c #ajj #d #g #ajj ----------------------1 ----------------------1 #f #c #d #g #ab #b #f #c #d #g #ab #b #ab #b #ajj & ----------------------4 #f #c #d #g #ab #b ,,ex]cises4 ,a1 ,b1 & ,c emb>k 9 an und]tak+2 ,a plac+ #c4 #f = #b ye>s1 ,b #ajj = #a ye>1 & ,c #ab = #a ye>s4 ,!y ga9 #dbgf4 #g ,h[ mu* mu/ ea* receive ( ! ga98 ,answ]1 ,a #bbf4 #aj4 #d2 ,b #cdcb4 #a4 #c2 ,c #fag4 #ae4 #e4 ,a1 ,b1 & ,c r5t a h\se toge!r = #b ye>s1 at #aej p] annum4 ,a rema9s 9 x ! :ole "t1 ,b #af mon?s1 & ,c #d mon?s1 dur+ ! occupancy ( ,b4 ,h[ mu* mu/ ea* pay ( ! r5t8#ef ,answ]1 ,a %\ld pay #aij4 #ab4 #f2 ,b #ij4 #ab4 #f2 ,c #ah4 #ae4 #ef ,?is "q does n at fir/ appe> to fall "u ! rule4 ,a little ?\ght w s]ve to %ew t :at probably w be ! fir/ idea ( ! prop] me?od ( solution is errone\s4 #beg4 ,^! are ! pr9cipal rules employ$ 9 ! application ( ari?metic to comm]ce4 ,!re are o!rs1 :i*1 z no "o :o und]/&s ! pr9ciples "h laid d[n c fail to see1 are virtually conta9$ 9 ^? :i* h be5 giv54 ,su* is :at is commonly call$ ! ,rule ( ,ex*ange1 = su* que/ions z ! foll[+3 ,if #bj %ill+s be wor? #be francs1 9 ,france1 :at is #afj wor?8 ,?is may evid5tly be done by ! ,rule ( ,?ree4 ,! rules "h giv5 are ^? :i* are mo/ useful 9 common life2 & ! /ud5t :o und]/&s !m ne$ n fe> t any ord9>y "q w be above 8 rea*4 ,b no /ud5t mu/ imag9e t f ?is or any o!r book ( ari?metic he w le>n precisely ! modes ( op]ation :i* are be/ adapt$ to ! wants ( ! p>ticul> k9d ( bus9ess 9 :i* 8 future life may be pass$4 ,!re is no su* ?+ z a set ( rules :i* are at once mo/ conv5i5t = a but*] & a bank]'s cl]k1 a groc] & an actu>y1 a f>m] & a bill-brok]2 b a p]son ) a good k ( ! pr9ciples laid d[n 9 ?is "w1 w be able to exam9e & meet 8 own future wants1 or1 at wor/1 to cat* ) read9ess ! mann] 9 :i* ^? :o h gone be=e him h done s = !mselves4 ,,app5dix ,,to ,,! ,,fif? ,,edition ,,( ,,de ,,morgan',s ,,elem5ts ,,( ,,ari?metic4 ,i4 ,,on ,,! ,,mode ,,( ,,comput+4 ,! rules 9 ! prec$+ "w are giv5 9 ! usual =m1 & ! examples are work$ 9 ! usual mann]4 ,b if ! /ud5t really wi% to become a ready comput]1 he %\ld /rictly foll[ ! me?ods laid d[n 9 ?is ,app5dix2 & he may dep5d ^u x t he w !reby save himself tr\ble 9 ! end1 z well z acquire habits ( quick & accurate calculation4 ,i4 ,9 num]ation le>n to connect ea* prim>y decimal numb]1 #aj1 #ajj1 #ajjj1 c4 n ) ! place 9 :i* ! unit falls1 b ) ! numb] ( ciph]s foll[+4 ,call t5 a "o-ciph] numb]1 a hundr$ a two-ciph] numb]1 a million a six-ciph] numb]1 & s on4 ,if five figures be cut (f f a numb]1 ^? t are left are hundr$-?\s&s2 = #ajj1jjj is a five-ciph] numb]4 ,le>n to connect t5s1 hundr$s1 ?\s&s1 t5s ( ?\s&s1 hundr$s ( ?\s&s1 millions1 c4 ) #a1 #b1 #c1 #d1 #e1 #f1 c4 9 ! m9d4 ,:at is a sev5te5-ciph] numb]8 ,= e #f 9 sev5te5 say million1 = ! rema9+ #e say hundr$-?\s&3 ! answ] is a hundr$ ?\s& millions ( millions4 ,if twelve places be cut (f f ! "r ( a numb]1 :at does ! rema9+ numb] /& =8--,answ]1 ,z _m millions ( millions z !re are units 9 x :5 /&+ by itself4 ,,ii4 ,aft] le>n+ to c\nt =w>ds & backw>ds ) rapidity1 z 9 #a1 #b1 #c1 #d1 c4 or #cj1 #bi1 #bh1 #bg1 c41 le>n to c\nt =w>ds or backw>ds by twos1 ?rees1 c4 up to n9es at lea/1 beg9n+ f any numb]4 ,?us1 beg9n+ f f\r & proce$+ by sev5s1 we h #d1 #aa1 #ah1 #be1 #cb1 c41 along :i* s]ies y mu/ le>n to g z easily z along ! s]ies #a1 #b1 #c1 #d1 c42 t is1 z quick z y c pron\nce ! words4 ,! act ( addition mu/ be made 9 ! m9d )\t assi/ance3 y mu/ n p]mit y\rself to say1 #d & #g are #aa1 #aa & #g are #ah1 c42 b only #d1 #aa1 #ah1 c4 ,& x w\ld be desirable1 ?\gh n s necess>y1 t y %\ld g back z readily z =w>d2 by sev5s = in/ance1 f sixty1 z 9 #fj1 #ec1 #df1 #ci1 c4 ,,iii4 ,see+ a numb] & ano!r bo? ( "o figure1 le>n to cat* in/antly ! numb] y mu/ add to ! small] to get ! great]4 ,see+ #c & #h1 le>n by practice to ?9k ( #e )\t ! necessity ( say+ #c f #h & !re rema9s #e4 ,& if ! second numb] be ! less1 z #h & #c1 le>n also by practice h[ to pass up f #h to ! next numb] :i* ends ) #c 7or #ac71 & to cat* ! necess>y augm5tation1 five1 )\t ! necessity ( =mally und]tak+ 9 words to subtract #h f #ac4 ,take r[s ( numb]s1 su* z #d #b #f #j #e #j #a #h #f #d & practise ?is rule ^u e figure & ! next1 n p]mitt+ y\rself 9 ?is simple case ev] to "n ! high] "o4 ,?us1 say #d & #h 7#d fir/1 #b second1 #d f ! next numb] t ends ) #b1 or #ab1 leaves #h71 #b & #d1 #f & #d1 #j & #e1 #e & #e1 #j & #a1 #a & #g1 #h & #h1 #f & #h4 ,,iv4 ,/udy ! same ex]cise z ! la/ "o ) two figures & "o4 ,?us1 see+ #bg & #f1 pass f #bg up to ! next numb] t ends ) #f 7or #cf71 cat* ! #i ?r\gh :i* y h to pass1 & all[ y\rself to repeat z mu* z 8#bg & #i are #cf40 ,?us1 ! r[ ( figures #aggbifchaji w give ! foll[+ practice3 #ag & #j are #ag2 #gg & #e are #hb2 #gb & #g are #gi2 #bi & #g are #cf2 #if & #g are #ajc2 #fc & #e are #fh2 #ch & #c are #da2 #ha & #i are #ij2 #aj & #i are #ai4 ,v4 ,9 a numb] ( two figures1 practise writ+ d[n ! units at ! mom5t t y are keep+ ! att5tion fix$ ^u ! t5s4 ,9 ! prec$+ ex]cise1 = in/ance1 write d[n ! results1 repeat+ ! t5s ) emphasis at ! in/ant ( writ+ d[n ! units4 ,,vi4 ,le>n ! multiplication table s well z to "n ! product ! in/ant ! factors are se52 t is1 until #h & #g1 or #g & #h1 sugge/ #ef at once1 )\t ! necessity ( say+ 8#g times #h are #ef40 ,?us look+ along a r[ ( numb]s1 z #cigjfedh1 le>n to "n ! products ( e successive pair ( digits z fa/ z y c repeat !m1 namely1 #bg1 #fc1 #j1 #j1 #cj1 #bj1 #cb4 ,,vii4 ,hav+ ?or\ghly ma/]$ ! la/ ex]cise1 le>n fur!r1 on see+ ?ree numb]s1 to augm5t ! product ( ! fir/ & second by ! ?ird )\t any repetition ( words4 ,practise until #c1 #h1 #d1 = in/ance1 sugge/ #c times #h & #d1 or #bh1 )\t ! necessity ( say+ 8#c times #h are #bd1 & #d is #bh40 ,?us1 #agibcfdjh w sugge/ ! foll[+ practice1 #af1 #fe1 #ba1 #ab1 #bb1 #bd1 #h4 ,,viii4 ,n[1 c>ry ! la/ /ill fur!r1 z foll[s3 ,see+ f\r figures1 z #b1 #g1 #f1 #i1 cat* up ! product ( ! fir/ & second1 increas$ by ! ?ird1 z 9 ! la/1 )\t a help+ ^w2 "n ! result1 & add ! next figure1 "n ! :ole result1 lay+ emphasis ^u ! t5s4 ,?us1 #b1 #g1 #f1 #i1 mu/ imm$iately sugge/ 8#bj & #i are #bi40 ,! r[ ( figures #ggcfihigd w give ! in/ances #eb & #f are #eh2 #bg & #i are #cf2 #bg & #h are #ce2 #fb & #i are #ga2 #ha & #g are #hh2 #gi & #d are #hc4 ,,ix4 ,hav+ f\r numb]s1 z #b1 #d1 #g1 #i1 v>y ! la/ ex]cise z foll[s3 ,cat* ! product ( ! fir/ & second1 increas$ by ! ?ird2 b in/ead ( add+ ! f\r?1 g up to ! next numb] t ends ) ! f\r?1 z 9 ex]cise ,,iv4 ,?us1 #b1 #d1 #g1 #i1 are to sugge/ 8#ae & #d are #ai40 ,& ! r[ ( figures #agbcifhibi w af=d ! in/ances #i & #d are #ac2 #ag & #b are #ai2 #ae & #a are #af2 #cc & #e are #ch2 #fb & #g are #fi2 #eg & #e are #fb2 #gd & #e are #gi4 ,x4 ,le>n to f9d rapidly ! numb] ( times a digit is conta9$ 9 giv5 units & t5s1 ) ! rema9d]4 ,?us1 see+ #h & #ec1 arrive at & repeat 8#f & #e ov]40 ,common %ort division is ! be/ practice4 ,?us1 9 divid+ #bcfdajgib by #g1 #g7bcfdajgib --------- #ccggbigj1 rema9d] #b4 ,all t is repeat$ %\ld be #c & #b2 #c & #e2 #g & #e2 #g & #b2 #b & #f2 #i & #d2 #g & #j2 #j & #b4 ,9 p]=m+ ! sev]al rules1 proce$ z foll[s3 ,,addition4 ,n "o ^w m ?an repeat+ ! numb]s writt5 9 ! foll[+ process3 ! acc5t$ figure is ! "o to be writt5 d[n2 ! d\bly acc5t$ figure is c>ri$ 7& don't say 8c>ry #c10 b d x74 #dgifc #aeih #bfcaf #edgib #hai #ffhf ------ #achagd #f1 #ae1 #ag1 #bc1 #ca1 #c #d2 #aa1 #ab1 #ba1 #bb1 #ca1 #cg2 #i1 #ag1 #bd1 #bg1 #cb1 #da2 #aj1 #ad1 #bj1 #ba1 #bh2 #g1 #i1 #ac4 ,9 v]ify+ additions1 in/ead ( ! usual way ( omitt+ "o l9e1 add+ )\t x1 & !n add+ ! l9e omitt$1 v]ify ea* column by add+ x bo? upw>ds & d[nw>ds4 ,,subtraction4 ,! foll[+ process is en\gh4 ,! c>riages1 be+ always ( "o1 ne$ n be m5tion$4 ,f #gidcfbehaij ,take #ehfdeifbgch ----------- #bjgijbiedeb #h & #b1 #d & #e1 #g & #d1 #c & #e1 #f & #i1 #aj & #b1 #f & #j1 #d & #i1 #g & #g1 #i & #j1 #e & #b4 ,x is useless to /op & say1 #h & #b make #aj2 = z soon z ! #b is obta9$1 !re is no occasion to rememb] :at x came f4 ,,multiplication4 ,! foll[+1 put into words1 is all t ne$ be repeat$ 9 ! multiply+ "p2 ! addition is !n done z usual4 ,! unacc5t$ figures are c>ri$4 #fgjchc #ihgf ------- #djbbbih #ah1 #di1 #bb1 #b1 #db1 #dj1 #dfibfha #ba1 #eh1 #bf1 #b1 #di1 #df1 #ecfcjfd #bd1 #ff1 #cj1 #c1 #ef1 #ec1 #fjccddg #bg1 #gd1 #cd1 #c1 #fc1 #fj4 ---------- #ffbjgjbejh ,v]ify ea* l9e ( ! multiplication & ! f9al result by ca/+ out ! n9es4 7,app5dix ,,ii4 p4 #aff47 ,x w\ld be almo/ z easy1 = a p]son :o has well practis$ ! #h? ex]cise1 to add ea* l9e to ! "o be=e 9 ! process1 ?us3 #fgjchc #ihgf ------- #djbbbih #ejidiajh #ehgbeeejh #ffbjgjbejh #h2 #ba & #i are #cj2 #ei & #b are #fa2 #bg & #b are #bi2 #b & #b are #d2 #di & #j are #di2 #df & #d are #ej4 ,on ! "r is all ! process ( =m+ ! second l9e1 :i* completes ! multiplication by #gf1 z ! ?ird l9e completes t by #hgf1 & ! f\r? l9e t by #ihgf4 ,,division4 ,make ea* multiplication & ! foll[+ subtraction 9 "o /ep1 by help ( ! process 9 ! #i? ex]cise1 z foll[s3 #bgfic7ddaigbhjiffb7aeieigcj #afejdb #bfeggh #afedaj #bfidei #bjbbbf #hcgef #fggb ,! numb] ( words by :i* #bfegg is obta9$ f #afedjb 7! multipli] be+ #e7 is z foll[s3 #ae & #g are #bb2 #dg & #g are #ed2 #ce & #e are #dj2 #ci & #f are #de2 #ad & #b are #af4 ,! processes = extract+ ! squ>e root1 & = ! solution ( equations 7,app5dix ,,xi471 %\ld be abbreviat$ 9 ! same mann] z ! division4#eg #eg ,! tea*] w f9d fur!r rem>ks on ?is subject 9 ! ,companion to ! ,almanac = #ahdd1 & 9 ! ,supplem5t to ! ,p5ny ,cyclopdia1 article ,computation4 ,,app5dix ,,ii4 ,,on ,,v]ification ,,by ,,ca/+ ,,out ,,n9es ,,& ,,elev5s4 ,! process ( ca/+ out ! n9es1 z x is call$1 is "o :i* ! "y comput] %\ld le>n & practise1 z a *eck ^u 8 computations4 ,x is n a complete *eck1 s9ce if "o figure 7 made too small1 & ano!r z mu* too great1 x w\ld n detect ?is d\ble error2 b z x is v unlikely t su* a d\ble error %\ld take place1 ! *eck furni%es a /rong presumption ( accuracy4 ,! proposition ^u :i* ?is me?od dep5ds is ! foll[+3 ,if a1 b1 c1 d be f\r numb]s1 su* t a bc d1 & if m be any o!r numb] :atsoev]1 & if a1 b1 c1 d1 sev]ally divid$ by m1 give ! rema9d]s p1 q1 r1 s1 !n p & qr s give ! same rema9d] :5 divid$ by m 7& p]haps are !mselves equal74 ,= in/ance1 #ccd #ag #ai #aa2 divide ^! f\r numb]s by #g1 ! rema9d]s are #e1 #c1 #e1 & #d4 ,& #e & #e #c #d1 or #e & #ai1 bo? leave ! rema9d] #e :5 divid$ by #g4 ,any numb]1 !re=e1 be+ us$ z a divisor1 may be made a *eck ^u ! correctness ( an op]ation4 ,to provide a *eck :i* may be mo/ fit = use1 we mu/ take a divisor ! rema9d] to :i* is mo/ easily f\nd4 ,! mo/ conv5i5t divisors are #c1 #i1 & #aa1 ( :i* #i is f> ! mo/ useful4 ,z to ! numb]s #c & #i1 ! rema9d] is always ! same z t ( ! sum ( ! digits4 ,= in/ance1 requir$ ! rema9d] ( #bdfabjcgg divid$ by #i4 ,! sum ( ! digits is #b #d #f #a #b #j #c #g #g1 or #cb1 :i* gives ! rema9d] #e4 ,b ! easie/ way ( proce$+ is by ?r[+ out n9es z fa/ z !y arise 9 ! sum4 ,?us1 repeat #b1 #f 7#b #d71 #ab 7#f #f71 say #c 7?r[+ out #i71 #d1 #f1 #i 7?r[ ?is away71 #g1 #ad1 7or ?r[+ out ! #i7 #e4 ,?is is ! rema9d] requir$1 z w\ld appe> by divid+ #bdfabjcgg by #i4 ,a pro( may be giv5 ?us3 ,x is obvi\s t ea* ( ! numb]s1 #a1 #aj1 #ajj1 #ajjj1 c4 divid$ by #i1 leaves a rema9d] #a1 s9ce !y are #a1 #i #a1 #ii #a1 c4 ,consequ5tly1 #b1 #bj1 #bjj1 c4 leave ! rema9d] #b2 #c1 #cj1 #cjj1 ! rema9d] #c2 & s on4 ,if1 !n1 we divide1 say #agfd by #i 9 p>cels1 #ajjj w be "o m ?an an exact numb] ( n9es1 #gjj w be sev5 m1 & #fj w be six m4 ,s1 !n1 f #a1 #g1 #f1 #d1 put toge!r1 & ! n9es tak5 out1 comes ! only rema9d] :i* c come f #agfd4 ,to apply ?is process to a multiplication3 ,x is ass]t$1 9 page #cb1 t #ajjjdefi #cafc #cafdddeagdg4 ,9 ca/+ out ! n9es f ! fir/1 all t is necess>y to repeat is1 "o1 five1 t51 "o1 sev52 9 ! second1 ?ree1 f\r1 t51 "o1 f\r2 9 ! ?ird1 ?ree1 f\r1 t51 "o1 five1 n9e1 f\r1 n9e1 eight1 twelve1 ?ree1 t51 "o4 ,! rema9d]s !n are1 #g1 #d1 #a4 ,n[1 #g #d is #bh1 :i*1 ca/+ out ! n9es1 gives #a1 ! same z ! product4 ,aga91 9 page #dc1 x is ass]t$ t #bcgifdhd #acjjjj #ahc #fdhd4 ,ca/ out ! n9es f #acjjj1 #ahc1 #fdhd1 & we h #d1 #c1 & #d4 ,n[1 #d #c #d1 ) ! n9es ca/ out1 gives #g2 & s does #bcgifdhd4 ,to avoid hav+ to rememb] ! result ( "o side ( ! equation1 or to write x d[n1 9 ord] to confront x ) ! result ( ! o!r side1 proce$ z foll[s3 ,hav+ got ! rema9d] ( ! m complicat$ side1 into :i* two or m numb]s ent]1 subtract x f #i1 & c>ry ! rema9d] into ! simple side1 9 :i* !re is only "o numb]4 ,!n ! rema9d] ( t side ought to be #j4 ,?us1 hav+ got #g f ! left-h& ( ! prec$+1 take #b1 ! re/ ( #i1 =get #g1 & c>ry 9 #b z a beg9n+ to ! left-h& side1 giv+ #b1 #d1 #g1 #ad1 #e1 #aa1 #b1 #f1 #ad1 #e1 #i1 #j4 ,practice w enable ! /ud5t to ca/ out n9es ) great rapidity4 ,?is process ( ca/+ out ! n9es does n detect any errors 9 :i* ! rema9d] to #i happ5s to be correct4 ,if a process be t$i\s1 & "s additional *eck be desirable1 ! me?od ( ca/+ out elev5s may be foll[$ aft] t ( ca/+ out ! n9es4 ,obs]ve t #aj #a1 #ajj-a1 #ajjj #a1 #ajjjj-a1 c4 are all divisible by elev54 ,f ?is ! foll[+ rule = ! rema9d] ( division by #aa may be d$uc$1 & readily us$ by ^? :o "k ! algebraical process ( subtraction4 ,= ^? :o h n got s f>1 x may be d\bt$ :e!r ! rule c be made easi] ?an ! actual division by #aa4 ,subtract ! fir/ figure f ! second1 ! result f ! ?ird1 ! result f ! f\r?1 & s on4 ,! f9al result1 or ! re/ ( #aa if ! figure be negative1 is ! rema9d] requir$4 ,?us1 to divide #afdbiae by #aa1 & f9d ! rema9d]1 we h #a f #f1 #e2 #e f #d1-a2-a f #b1 #c2 #c f #i1 #f2 #f f #a1-e2-e f #e1 #aj2 & #aj is ! rema9d]4 ,b #afd gives-#a1 & #aj is ! rema9d]2 #afdbia gives-#e1 & #f is ! rema9d]4 ,) v little practice ^! rema9d]s may be read z rapidly z ! numb] itself4 ,?us1 = #abgfaihccdbd ne$ only be repeat$1 #a1 #f1 #j1 #a1 #h1 #j1 #c1 #j1 #d1-b1 #f1 & #f is ! rema9d]4 ,:5 a "q has be5 tri$ bo? by n9es & elev5s1 !re c be no error unless x be "o :i* makes ! result wrong by a numb] ( times #ii exactly4 ,,app5dix ,,iii4 ,,on ,,scales ,,( ,,notation4 ,we are s well accu/om$ to #aj1 #ajj1 c41 z /&+ = t51 t5 t5s1 c41 t we are n apt to rememb] t !re is no reason :y #aj might n /& = five1 #ajj = five fives1 c41 or = twelve1 twelve twelves1 c4 ,because we inv5t diff]5t columns ( numb]s1 & let units 9 ! diff]5t columns /& = collections ( ! units 9 ! prec$+ columns1 we are n !re=e b\nd to all[ ( no collections except 9 t5s4 ,if #aj /ood = #b1 t is1 if e column _h its unit d\ble ( ! unit 9 ! column on ! "r1 :at we n[ repres5t by #a1 #b1 #c1 #d1 #e1 #f1 c41 w\ld be repres5t$ by #a1 #aj1 #aa1 #ajj1 #aja1 #aaj1 #aaa1 #ajjj1 #ajja1 #ajaj1 #ajaa1 #aajj1 c4 ,?is is ! b9>y scale4 ,if we take ! t]n>y scale1 9 :i* #aj /&s = #c1 we h #a1 #b1 #aj1 #aa1 #ab1 #bj1 #ba1 #bb1 #ajj1 #aja1 #ajb1 #aaj1 c4 ,9 ! qu9>y scale1 9 :i* #aj is five1 #bcd /&s = #b tw5ty-fives1 #c fives1 & #d1 or sixty-n9e4 ,if we take ! duod5>y scale1 9 :i* #aj is twelve1 we mu/ inv5t new symbols = t5 & elev51 because #aj & #aa n[ /& = twelve & ?irte52 use ! lett]s t & e4 ,!n #agf means #a twelve-twelves1 #g twelves1 & #f1 or two hundr$ & ?irty-f\r2 & #ate means two hundr$ & sev5ty-five4 ,! numb] :i* #aj /&s = is call$ ! radix ( ! scale ( notation4 ,to *ange a numb] f "o scale into ano!r1 divide ! numb]1 writt5 z 9 ! fir/ scale1 by ! numb] :i* is to be ! radix ( ! new scale2 repeat ?is division aga9 & aga91 & ! rema9d]s are ! digits requir$4 ,= example1 :at1 9 ! qu9>y scale1 is t numb] :i*1 9 ! decimal scale1 is #agjcf8 #e7agjcf ----- #e7cdjg ,rem4 #a ---- #e7fha #b --- #e7acf #a --- #e7bg #a -- #e7e #b - #e7a #j - #j #a ,answ] #ajbaaba ,qu9>y4 ,decimal4 ,v]ification1 #ajjjjjj means #aefbe #bjjjj #abej #ajjj #abe #ajj #be #bj #aj #a #a ------ ----- #ajbaaba #agjcf ,! reason ( ?is rule is easy4 ,our process ( division is no?+ b tell+ (f #agjcf into #cdjg fives & #a ov]2 we !n f9d #cdjg fives to be #fha fives ( fives & #b fives ov]4 ,next we =m #fha fives ( fives into #acf fives ( fives ( fives & #a five ( fives ov]2 & s on4 ,x is a useful ex]cise to multiply & divide numb]s repres5t$ 9 o!r scales ( notation ?an ! common or decimal "o4 ,! rules are 9 all respects ! same = all sy/ems1 ! numb] c>ri$ be+ always ! radix ( ! sy/em4 ,?us1 9 ! qu9>y sy/em we c>ry fives in/ead ( t5s4 ,i n[ give an example ( multiplication & division3 ,qu9>y4 ,decimal4 #dbadc means #bgih #abcd #aid ------ ----- #cbdbcb #aaaib #bcbjcd #beahb #acdcda #bgih #dbadc --------- ------ #aadccbbbb #edbhab ,duodecimal4 ,decimal4 #dt#i7gft#d;e#jh7affhg #gje7bbfajgdd7cbjga #dt#i #adfj ----- #ejgd #bhad #acid #bedf #fhi ----- #bhte #bedf ------ #cfej #ccbj ----- #ccjh #bt#cc ------ #die ,ano!r way ( turn+ a numb] f "o scale into ano!r is z foll[s3 ,multiply ! fir/ digit by ! old radix 9 ! new scale1 & add ! next digit2 multiply ! result aga9 by ! old radix 9 ! new scale1 & take 9 ! next digit1 & s on to ! end1 always us+ ! radix ( ! scale y want to leave1 & ! notation ( ! scale y want to end 94 ,?us1 suppose x requir$ to turn #affhg 7duodecimal7 into ! decimal scale1 & #afdcb 7sept5>y7 into ! quat]n>y scale3 #affhg #afdcb ,duodecimals into ,decimals4 ,sept5>ies into ,quat]n>ies4 #a #ab #f #ah #a #g #f #ca #ab #f #g #d --- ---- #bbb #aacc #ab #h #g #c ---- ----- #bfgb #bbacj #ab #g #g #b ------ ------- ,answ] #cbjga #ajbajab ,ow+ to our division ( a foot into #ab equal p>ts1 ! duodecimal scale (t5 becomes v conv5i5t4 ,let ! squ>e foot be also divid$ into #ab p>ts1 ea* "p is #ab squ>e in*es1 & ! #ab? ( ! #ab? is "o squ>e in*4 ,suppose1 n[1 t ! two sides ( an oblong piece ( gr\nd are #agf feet #i in*es #g-ab?s ( an in*1 & #fe feet #aa in*es #e-ab?s ( an in*4 ,us+ ! duodecimal scale1 & duodecimal fractions1 ^! numb]s are #abhig & #ee;e#e4 ,_! product1 ! numb] ( squ>e feet requir$1 is ?us f\nd3 #abhig #ee;e#e --------- #fag;ee #aafjie #fag;ee #fag;ee ------------ #fh;e#hadd;e ,answ]1 #fh;e#hadd;e 7duod47 squ>e feet1 or #aaffj squ>e feet #af squ>e in*es _/ & _/ ( a squ>e in*4 ,x w\ld1 h[ev]1 be exact en\gh to all[ #b-;hundr$?s ( a foot = e qu>t] ( an in*1 an additional hundr$? = e #c in*es1#eh & #a-;hundr$? m if !re be a #ab? or #b-ab?s above ! qu>t] ( an in*4 ,?us1 #i_/ in*es %\ld be #gf #jc #ja1 or #hj1 & #aa_/ w\ld be #ie2 & ! prec$+ might !n be f\nd decimally z #agfh #feie z #aafeiif squ>e feet1 ne> en\gh = e practical purpose4 #eh ,& at discretion "o hundr$? m = a l>ge fraction ( ?ree in*es4 ,,app5dix ,,iv4 ,,on ,,! ,,def9ition ,,( ,,fractions4 ,! def9ition ( a fraction giv5 9 ! text %ews t _/1 = in/ance1 is ! n9? "p ( sev51 :i* is %ewn to be ! same ?+ z sev5-n9?s ( a unit4 ,b !re are v>i\s modes ( spee* "u :i* a fraction may be signifi$1 all ( :i* are m or less 9 use4 #a4 ,9 _/ we h ! #i? "p ( #g4 #b4 #g-i?s ( a unit4 #c4 ,! fraction :i* #g is ( #i4 #d4 ,! times & p>ts ( a "t 79 ?is case "p ( a "t only7 :i* #g conta9s #i4 #e4 ,! multipli] :i* turns n9es into sev5s4 #f4 ,! ratio ( #g to #i1 or ! proportion ( #g to #i4 #g4 ,! multipli] :i* alt]s a numb] 9 ! ratio ( #i to #g4 #h4 ,! #d? proportional to #i1 #a1 & #g4 ,! fir/ two views are 9 ! text4 ,! ?ird is d$uc$ ?us3 ,if we divide #i into #i equal p>ts1 ea* is #a1 & #g ( ! p>ts are #g2 consequ5tly ! fraction :i* #g is ( #i is _/4 ,! f\r? view foll[s imm$iately3 ,= a "t is only a ^w us$ to express "o ( ! repetitions :i* take place 9 multiplication1 & we all[ ourselves1 by an easy ext5sion ( language1 to speak ( a portion ( a numb] z be+ t numb] tak5 a "p ( a "t4 ,! fif? view is no?+ m ?an a *ange ( words3 ,a numb] r$uc$ to _/ ( its am\nt has e #i conv]t$ into a #g1 & any fraction ( a #i :i* may rema9 ov] into ! correspond+ fraction ( #g4 ,?is is completely prov$ :5 we prove ! equation _/ ( a #g times a_/#i4 ,! six?1 sev5?1 & eigh? views are illu/rat$ 9 ! *apt] on proportion4 ,:5 ! /ud5t comes to algebra1 he w f9d t1 9 all ! applications ( t sci5ce1 fractions su* z a_/b mo/ frequ5tly require t a & b %\ld be !mselves suppos$ to be fractions4 ,x is1 !re=e1 ( importance t he %\ld le>n to accommodate 8 views ( a fraction to ?is m complicat$ case4 #b ,suppose we take -----4 #d_/ ,we %all f9d t we h1 9 ?is case1 a bett] idea ( ! views f & aft] ! ?ird inclusive1 ?an ( ! fir/ & second1 :i* are c]ta9ly ! mo/ simple ways ( conceiv+ _/4 ,we h no notion ( ! 7#d_/7? "p ( #b1 #a 7 #c 7 nor ( #b ---7#d---7?s #b 7 #e 7 ( a unit2 inde$1 we co9 a new species ( adjective :5 we talk ( ! 7#d_/7? "p ( any?+4 ,b we c readily imag9e t #b is "s fraction ( #d_/2 t ! fir/ is "s "p ( a "t ! second2 t !re mu/ be "s multipli] :i* turns e #d_/ 9 a numb] into #b2 & s on4 ,let u n[ see :e!r we c inv5t a di/9ct mode ( apply+ ! fir/ & second views to su* a comp\nd fraction z ! above4 ,we c easily imag9e a f\r? "p ( a l5g?1 & a fif? "p1 mean+ ! l9es ( :i* #d & #e make up ! l5g? 9 "q2 & !re is also 9 exi/5ce a l5g? ( :i* f\r l5g?s & two-fif?s ( a l5g? make up ! orig9al l5g? 9 "q4 ,= in/ance1 we might say t #f1 #f1 #b is a division ( #ad into #b equal p>ts--#b equal p>ts1 #f1 #f1 & a ?ird ( a "p1 #b4 ,s we might agree to say1 t ! 7#b7?1 or 7#b7rd1 or 7#b7/ 7! read] may co9 ! adjective z he pleases7 "p ( #ad is #f4 ,if we divide ! l9e ,a ,b into elev5 equal p>ts 9 ,c1 ,d1 ,e1 c41 we mu/ !n say t ,a ,c is ! #aa? "p1 ,illu/ration ,a ,d ! 7#e7?1 ,a ,e ! 7#c7?1 ,a ,f ! 7#b7?1 ,a ,g ! 7#b7?1 ,a ,h ! 7#a7?1 ,a ,i ! 7#a_/7?1 ,a ,k ! 7#a7?1 ,a ,l ! 7#a_/7?1 ,a ,m ! 7#a7?1 & ,a ,b itself ! #a/ "p ( ,a ,b4 ,! read] may refuse ! language if he likes 7?\gh x is n s mu* 9 defiance ( etymology z talk+ ( multiply+ by 72 b :5 ,a ,b is call$ #a1 he mu/ ei!r call ,a ,f #a_/7b71 or make "o def9ition ( "o class ( fractions & ano!r ( ano!r4 ,:atev] abbreviations !y may *oose1 all p]sons w agree t a_/b is a direction to f9d su* a fraction z1 repeat$ b times1 w give #a1 & !n to take t fraction a times4 ,s1 to get #b_/d1 ! simple/ way is to divide ! :ole unit into #df p>ts2 #aj ( ^! p>ts1 repeat$ #d times1 give ! :ole4 ,! ,illu/ration #d? is !n _/1 & #b su* p>ts is _/1 or ,a ,c4 ,! /ud5t %\ld try sev]al examples ( ?is mode ( int]pret+ complex fractions4 ,b :at are we to say :5 ! d5om9ator itself is less ?an unity1 z 9 #c_/8 ,are we to h a 77? "p ( a unit8 & :at is x8 ,_h !re be5 a #e 9 ! d5om9ator1 we %\ld h tak5 ! "p ( :i* #e w make a unit4 ,z !re is 9 ! d5om9ator1 we mu/ take ! "p ( :i* w be a unit4 ,t "p is l>g] ?an a unit2 x is #b units2 #b is t ( :i* is #a4 ,! above fraction !n directs u to repeat #b units #c times4 ,by ext5d+ our ^w #8multiplication' to ! tak+ ( a "p ( a "t1 all multiplications are also divisions1 & all divisions multiplications1 & all ! t]ms connect$ ) ei!r are subject to be appli$ to ! results ( ! o!r4 ,if #b y>ds co/ #c %ill+s1 h[ mu* does "o y>d co/8 ,9 su* a case z ?is1 ! /ud5t looks at a m simple "q4 ,if #e y>ds co/ #aj %ill+s1 he sees t ea* y>d co/s _/1 or #b %ill+s1 &1 conclud+ t ! same process w give ! true result :5 ! data are fractional1 he =ms #c_/b1 r$uces x by rules to _/ or #a1 & concludes t #a y>d co/s #ah p5ce4 ,! answ] happ5s to be correct2 b he is n to suppose t ?is rule ( copy+ = fractions :atev] is se5 to be true ( integ]s is "o :i* requires no demon/ration4 ,9 ! above "q we want money :i*1 repeat$ #b times1 %all give #c %ill+s4 ,if we divide ! %ill+ into #ad equal p>ts1 #f ( ^! p>ts repeat$ #b times give ! %ill+4 ,to get #c times z mu* by ! same repetition1 we mu/ take #c ( ^! #f p>ts at ea* /ep1 or #ba p>ts4 ,h5ce1 _/1 or #a1 is ! numb] ( %ill+s 9 ! price4 ,,app5dix ,v4 ,,on ,,*>act]i/ics4 ,:5 ! /ud5t comes to use log>i?ms1 he w f9d :at foll[s v useful4 ,9 ! mean :ile1 ,i give x m]ely z furni%+ a rapid rule = f9d+ ! place ( a decimal po9t 9 ! quoti5t be=e ! division is comm5c$4 ,:5 a b> is writt5 ov] a numb]1 ?us1 #g let ! numb] be call$ negative1 & let x be ?us us$3 ,let x be augm5t$ by additions ( its own species1 & dim9i%$ by subtractions2 ?us1 #g & #b give #i1 & let #g ) #b subtract$ give #e4 ,b let ! addition ( a numb] )\t ! b> dim9i% ! negative numb]1 & ! subtraction increase x4 ,?us1 #g & #d are #c1 #g & #ab make #e1 #g ) #h subtract$ is #ae4 ,9 fact1 consid] #a1 #b1 #c1 c41 z if !y 7 ga9s1 & #a1 #b1 #c1 z if !y 7 losses3 let ! addition ( a ga9 or ! removal ( a loss be equival5t ?+s1 & also ! removal ( a ga9 & ! addition ( a loss4 ,?us1 :5 we say t #d dim9i%$ by #aa gives #g1 we say t a loss ( #d incurr$ at ! mom5t :5 a loss ( #aa is remov$1 is1 on ! :ole1 equival5t to a ga9 ( #g2 & say+ t #d dim9i%$ by #b is #f1 we say t a loss ( #d1 accompani$ by ! removal ( a ga9 ( #b1 is altoge!r a loss ( #f4 ,by ! *>act]i/ic ( a numb] und]/& z foll[s3 ,:5 !re are places be=e ! decimal po9t1 x is "o less ?an ! numb] ( su* places4 ,?us1 #cbad1 #ajjhc1 #h 7:i* is #hjj 4447 #iiii1 all h #j = _! *>act]i/ics4 ,b #agcb1 #dh1 #icaaf1 all h #a2 #abfjc & #abf h #b2 #aaicgbfdfff has #g4 ,b :5 !re are no places be=e ! decimal po9t1 look at ! fir/ decimal place :i* is significant1 & make ! *>act]i/ic negative accord+ly4 ,?us1 #fab1 #aba1 #ijjd1 9 all ( :i* significance beg9s 9 ! fir/ decimal place1 h ! *>act]i/ic #a2 b #jah & #jii h #b2 #jjjag has #d2 #jjjjjjjja has #i4 ,to f9d ! *>act]i/ic ( a quoti5t1 subtract ! *>act]i/ic ( ! divisor f t ( ! divid5d1 c>ry+ "o be=e subtraction if ! fir/ significant figures ( ! divisor are great] ?an ^? ( ! divid5d4 ,= in/ance1 9 divid+ #adfjh by #jjbgi4 ,! *>act]i/ics are #b & #c2 & #b ) #c remov$ w\ld be #e4 ,b on look+1 we see t ! fir/ significant figures ( ! divisor1 #bg1 tak5 by !mselves1 & )\t ref]5ce to _! local value1 mean a l>g] numb] ?an #ad1 ! fir/ two figures ( ! divid5d4 ,consequ5tly1 to #c we c>ry #a be=e subtract+1 & x !n becomes #b1 :i*1 tak5 f #b1 gives #d4 ,& ?is #d is ! *>act]i/ic ( ! quoti5t1 s t ! quoti5t has #e places be=e ! decimal po9t4 ,or1 if abcdef be ! fir/ figures ( ! quoti5t1 ! decimal po9t mu/ be ?us plac$1 abcdef4 ,b if x _h be5 to divide #jjbgi by #adfjh1 no c>riage w\ld h be5 requir$2 & #c dim9i%$ by #b is #e2 t is1 ! fir/ significant figure ( ! quoti5t is 9 ! #e? place4 ,! quoti5t1 !n1 has #jjjj be=e any significant figure4 ,a few applications ( ?is rule w make x easy to d x 9 ! head1 & ?us to assign ! mean+ ( ! fir/ figure ( ! quoti5t ev5 be=e x is f\nd4 ,,app5dix ,,vi4 ,,on ,,decimal ,,money4 ,( all ! simplifications ( comm]cial ari?metic1 none is comp>able to t ( express+ %ill+s1 p5ce1 & f>?+s z decimals ( a p\nd4 ,! rules are !reby put almo/ ^u z good a foot+ z if ! c\ntry possess$ ! advantage ( a real decimal co9age4 ,any fraction ( a p\nd /]l+ may be decimalis$ by rules :i* c be made to give ! result at once4 ,two %ill+s is #ajj ,"o %ill+ is #jej ,sixp5ce is #jbe ,"o f>?+ is #jja #jd ,?us1 e pair ( %ill+s is a unit 9 ! fir/ decimal place2 an odd %ill+ is a #ej 9 ! second & ?ird places2 a f>?+ is s ne>ly ! ?\s&? "p ( a p\nd1 t to say "o f>?+ is #jja1 two f>?+s is #jjb1 c41 is s ne> ! tru? t x makes no error 9 ! fir/ ?ree decimals till we arrive at sixp5ce1 & !n #bd f>?+s is exactly #jbe or #be ?\s&?s4 ,b #be f>?+s is #jbf1 #bf f>?+s is #jbg1 c4 ,h5ce ! rule = ! fir/ ?ree places is ,"o 9 ! fir/ = e pair ( %ill+s2 #ej 9 ! second & ?ird = ! odd %ill+1 if any2 & #a = e f>?+ additional1 ) #a extra = sixp5ce4 ,?us1 #js4 #c;d4 #jad #js4 #g;d4 #jcb #as4 #b;d4 #jfj #as4 #aa;d4 #jif #bs4 #f;d4 #abe #bs4 #i;d4 #aci #cs4 #b;d4 #afa #acs4 #aj;d4 #fid ,9 ! f\r? & fif? places1 & ^? :i* foll[1 x is obvi\s t we h no produce f any f>?+s except ^? above sixp5ce4 ,= at e sixp5ce1 #jjjjd is conv]t$ into #jja1 & ?is has be5 already acc\nt$ =4 ,consequ5tly1 to fill up ! f\r? & fif? places1 ,take #d = e f>?+#ei above ! la/ sixp5ce1 & an additional #a = e six f>?+s1 or ?ree halfp5ce4 #ei ,! /ud5t %\ld rememb] all ! multiples ( #d up to #d #be1 or #ajj4 ,! rema9+ places arise altoge!r f #jjjjj = e f>?+ above ! la/ ?ree halfp5ce2 = at e ?ree halfp5ce complete1 #jjjjj is conv]t$ into #jjjja1 & has be5 already acc\nt$ =4 ,consequ5tly1 to fill up all ! places aft] ! fif?1 ,let ! numb] ( f>?+s above ! la/ ?ree halfp5ce be a num]ator1 #f a d5om9ator1 & annex ! figures ( ! correspond+ decimal fraction4 ,x may be easily rememb]$ t ,! figures ( _/ are #afffff444 0 _/ 444 #cccccc444 0 _/ 444 #e 0 _/ 444 #ffffff444 0 _/ 444 #hccccc444 #js4 #c;d4 #jadehcccc444 #js4 #g;d4 #jcbbiafff444 #as4 #b;d4 #jfjdaffff444 #as4 #aa;d4 #jifhge #bs4 #f;d4 #abejjjjjj444 #bs4 #i;d4 #aciehcccc444 #cs4 #b;d4 #afadehcccc444 #acs4 #aj;d4 #fidgiafff444 ,! foll[+ examples w %ew ! use ( ?is rule1 if ! /ud5t w also "w !m 9 ! common way4 ,to turn p\nds1 c41 into f>?+s3 ,multiply ! p\nds by #ifj1 or by #ajjj-dj1 or by #ajjj7a-_/72 t is1 f #ajjj times ! p\nds subtract #d p] c5t ( itself4 ,?us1 requir$ ! numb] ( f>?+s 9 #affc4 #aa4 #i4 #affceijfbe #ajjj #affceijfbe #d p] c5t ( ?is1 #ffedcfbe ----------- ,no4 ( f>?+s requir$1 #aeigjdg ,:at is #dg p] c5t ( #aff4 #ac4 #aj & #fadh ( #biga4 #af4 #i8 #afffia -------- #dj p4 c4 #fffgfd #e p4 c4 #hccdf #b p4 c4 #dafgc -------- #giaghc #gi4c4f #bigahcg --------- #f #aghcajbb #ja #bigahd #jjd #aahhgc #jjjh #bcgge --------- #ahbgjhed #ahbg4a4h ,! inv]se rule = turn+ ! decimal ( a p\nd into %ill+s1 p5ce1 & f>?+s1 is obvi\sly z foll[s3 ,a pair ( %ill+s = e unit 9 ! fir/ place2 an odd %ill+ = #ej 7if !re be #ej7 9 ! second & ?ird places2 & a f>?+ = e ?\s&? left1 aft] abat+ #a if ! numb] ( ?\s&?s s left exce$ #bd4 ,! direct rule 7) ?ree places7 gives too little1 ! inv]se rule too mu*1 except at ! end ( a sixp5ce1 :5 bo? are accurate4 ,?us1 #ahc is r less ?an #cs4 #h;d41 & #fs4 #d;d4 is r great] ?an #cai2 or :5 ! two d n exactly agree1 ! common money is ! greate/4 ,b #abe & #ce are exactly #bs4 #f;d4 & #gs4 ,requir$ ! price ( #ag cwt4 #ha lb4 #ac oz4 at #c4aa4i p] cwt4 true to ! hundr$? ( a f>?+4 #ceijfbe #ag --------- #fajdjfbe lb4 #ef #agiecac #af #eabidf #g #bbddad #b #jfdaah oz4 #h #jafjbi #d #jjhjae #a #jjbjjd #jjajjb --------- #fcffddff #fc4ac4c ,?ree m51 ,a1 ,b1 ,c1 sev]ally inve/ #aia4ab4g1 #fa4ad4h1 & #abb4a4i 9 an adv5ture :i* yields #eaa4ab4f4 ,h[ ought ! proce$s to be divid$ among !m8 ,a1 #aiafcbbi ,b1 #fagcccc ,c1 #abbjhieh ,produce ( #a4 --------- #cgedeebj7eaafbgjh7acfbfhf #acfagahh #bcececb #ajjhja #begaj #cahc #ahj #acfbfhf #acfbfhf #acfbfhf #ibbcfaia #ccccgaf #heihjbba --------- --------- --------- #acfbfhf #hagfab #acfbfhf #abbfdag #acfbg #bgbecg #acfbg #iech #bgbed #hagf #dji #ajij #dji #da #abb #bg #d #g #c -------- #a #a #hdabca --------- --------- #affcfig #bfaacdf #bfaacdf 444 ,a's %>e #bfa4b4h #hdabca 444 ,b's 444 #hd4b4e #affcfig 444 ,c's 444 #aff4g4d -------- ------------- #eaafbgd #eaa4b4f ,if ev] ! fraction ( a f>?+ be want$1 rememb] t ! co9age-result is l>g] ?an ! decimal ( a p\nd1 :5 we use only ?ree places4 ,f #ajjj times ! decimal take #d p] c5t1 & we get ! exact numb] ( f>?+s1 & we ne$ only look at ! decimal !n left to set ! prec$+ "r4 ,?us1 9 #acdf #abca #cfig #ech #dib #adgi ----- ------ ------ #bb #ah #ia we see t 7if we use f\r decimals only7 ! p5ce ( ! above results are ne>ly #h;d4 #bb ( a f>?+1 #e;d4 #ah1 & #d;d4 #ia4 ,a man c pay #bcgf4 #d4 #d1 8 debts be+ #cbic4 #aa4 #j4 ,h[ mu* p] c5t c he pay1 & h[ mu* 9 ! p\nd8 #cbiceec7bcgfbahj7gbadgef #gjgcji #dheih #aeffb #bdhh #ahc #ah ,answ]1 #gb4 #b4aa p] c5t4 #j4ad4e p] p\nd4 ,,app5dix ,,vii4 ,,on ,,! ,,ma9 ,,pr9ciple ,,( ,,book-,,keep+4 ,a brief notice ( ! pr9ciple on :i* acc\nts are kept 7:5 !y are prop]ly kept7 may p]haps be useful to /ud5ts :o are le>n+ book-keep+1 z ! treatises on t subject frequ5tly give too little 9 ! way ( explanation4 ,any p]son :o is engag$ 9 bus9ess mu/ desire to "k accurately1 :5ev] an inve/igation ( ! /ate ( 8 affairs is made4 #a1 ,:at he _h at ! comm5cem5t ( ! acc\nt1 or imm$iately aft] ! la/ inve/igation 0 made2 #b1 ,:at he has ga9$ & lo/ 9 ! int]val 9 all ! sev]al bran*es ( 8 bus9ess2 #c1 ,:at he is n[ wor?4 ,f ! fir/ two ( ^! ?+s he obvi\sly kn[s ! ?ird4 ,9 ! int]val betwe5 two inve/igations1 he may at any "o "t desire to "k h[ any "o acc\nt /&s4 ,an acc\nt is a recital ( all t has happ5$1 9 ref]5ce to any class ( deal+s1 s9ce ! la/ inve/igation4 ,x c only consi/ ( receipts & exp5ditures1 & s x is said to h two sides1 a debtor & a cr$itor side4 ,all acc\nts are kept 9 money4 ,if goods be b\ght1 !y are e/imat$ by ! money paid = !m4 ,if a debtor give a bill ( ex*ange1 be+ a promise to pay a c]ta9 sum at a c]ta9 "t1 x is put d[n z wor? t sum ( money4 ,all ! tools1 furniture1 horses1 c4 us$ 9 ! bus9ess are rat$ at _! value 9 money4 ,all ! actual co91 bank-notes1 c41 :i* are 9 or come 91 be+ ! only money 9 ! books :i* really is money1 is call$ ca%4 ,! acc\nts are kept z if e diff]5t sort ( acc\nt belong$ to a sep>ate p]son1 & _h an int]e/ ( its own1 :i* e transaction ei!r promotes or injures4 ,if ! /ud5t f9d t x helps him1 he may imag9e a cl]k to e acc\nt3 "o to take *>ge (1 & regulate1 ! actual ca%2 ano!r = ! bills :i* ! h\se is to receive :5 due2 ano!r = ^? :i* x is to pay :5 due2 ano!r = ! clo? 7if ! conc]n deal 9 clo?72 ano!r = ! sug> 7if x deal 9 sug>72 "o = e p]son :o has an acc\nt ) ! h\se2 "o = ! pr(its & losses2 & s on4 ,all ^! cl]ks 7or acc\nts7 belong+ to "o m]*ant1 mu/ acc\nt to him 9 ! end--mu/ ei!r produce all !y h tak5 9 *>ge1 or relieve !mselves by %ew+ to :om x w5t4 ,= all t !y h receiv$1 = e responsibility !y h und]tak5 to ! conc]n itself1 !y are b\nd1 or are debtors2 = ev]y?+ :i* has pass$ out ( *>ge1 or ab\t :i* !y are reliev$ f answ]+ to ! conc]n1 !y are unb\nd1 or are cr$itors4 ,^! words mu/ be tak5 9 a v wide s5se by any "o to :om book-keep+ is n to be a my/]y4 ,?us1 :5ev] any acc\nt assumes responsibility to any p>ties out ( ! conc]n1 x mu/ be cr$itor 9 ! books1 & debtor :5ev] x dis*>ges any o!r p>ties ( _! responsibility4 ,b :5ev] an acc\nt removes responsibility f any o!r acc\nt 9 ! same books x is debtor1 & cr$itor :5ev] x imposes ! same4 ,to :om are all ^! p>ties1 or acc\nts1 b\nd1 & f :om are !y releas$8 ,und\bt$ly ! m]*ant himself1 or1 m prop]ly1 ! balance-cl]k1 pres5tly m5tion$4 ,b x is cu/om>y to say t ! acc\nts are debtors to ea* o!r1 & cr$itors by ea* o!r4 ,?us1 ca% debtor to bills receivable1 means t ! ca% acc\nt 7or ! cl]k :o keeps x7 is b\nd to answ] = a sum :i* 0 paid on a bill ( ex*ange due to ! h\se4 ,at full l5g? x w\ld be3 8,mr4 ,c 7:o keeps ! ca%-box7 has receiv$1 & is answ]able =1 ?is sum :i* has be5 paid 9 by ,mr4 ,a1 :5 he paid 8 bill ( ex*ange40 ,on ! o!r h&1 ! correspond+ entry 9 ! acc\nt ( bills receivable runs--bills receivable1 cr$itor by ca%4 ,at full l5g?3 8,mr4 ,b 7:o keeps ! bills receivable7 is fre$ f all responsibility = ,mr4 ,a's bill1 :i* he once held1 by h&+ ov] to ,mr4 ,c1 ! ca%-cl]k1 ! money ) :i* ,mr4 ,a took x up40 ,bills receivable cr$itor by ca% is intelligible1 b ca% debtor to bills receivable is a misnom]4 ,! ca% acc\nt is debtor to ! m]*ant by ! sum receiv$ = ! bill1 & x %\ld be ca% debtor by bill receivable4 ,! fiction ( debts1 n "o ( :i* is ev] paid to ! p>ty to :om x is said to be ow+1 ?\gh ( no consequ5ce 9 practice1 is a /umbl+-block to ! le>n]2 b he mu/ keep ! phrase1 & rememb] its true mean+4 ,! acc\nt :i* is made debtor1 or b\nd1 is said to be debit$2 t :i* is made cr$itor1 or releas$1 is said to be cr$it$4 ,all :o receive mu/ be debit$2 all :o give mu/ be cr$it$4 ,no cancel is ev] made4 ,if ca% receiv$ be aft]w>ds repaid1 ! sum paid is n /ruck (f ! receipts 7or debtor-side ( ! ca% acc\nt71 b a dis*>ge1 or cr$it1 is writt5 on ! exp5diture 7or cr$it7 side4 ,! book 9 :i* ! acc\nts are kept is call$ a l$g]4 ,x has d\ble columns1 or else ! debtor-side is on "o page1 & ! cr$itor side on ! opposite1 ( ea* acc\nt4 ,! debtor-side is always ! left4 ,o!r books are us$1 b !y are only to help 9 keep+ ! l$g] correct4 ,?us !re may be a wa/e-book1 9 :i* all transactions are ent]$ z !y occur1 9 common language2 a j\rnal1 9 :i* ! transactions describ$ 9 ! wa/e-book are ent]$ at /at$ p]iods1 9 ! language ( ! l$g]4 ,! items ent]$ 9 ! j\rnal h ref]5ces to ! pages ( ! l$g] to :i* !y are c>ri$1 & ! items 9 ! l$g] h also ref]5ces to ! pages ( ! j\rnal f :i* !y come2 & by ?is mode ( ref]5ce x is easy to make a great deal ( abbreviation 9 ! l$g]4 ,?us1 :5 x happ5s1 9 mak+ up ! j\rnal to a c]ta9 date1 t sev]al diff]5t sums 7 paid or receiv$ at or ne> ! same "t1 ! totals may be ent]$ 9 ! l$g]1 & ! ca% acc\nt may be made debtor to1 or cr$itor by1 sundry acc\nts1 or sundries2 ! sundry acc\nts be+ sev]ally cr$it$ or debit$ = _! %>es ( ! :ole4 ,! only book t ne$ be expla9$ is ! l$g]4 ,all ! o!r books1 & ! mann] 9 :i* !y are kept1 important z !y may be1 h no?+ to d ) ! ma9 pr9ciple ( ! me?od4 ,let u1 !n1 suppose t all ! items are ent]$ at once 9 ! l$g] z !y arise4 ,x has appe>$ t e item is ent]$ twice4 ,if ,a pay on acc\nt ( ,b1 !re is an entry1 8,a1 cr$itor by ,b20 & ano!r1 8,b1 debtor to ,a40 ,?is is :at is call$ d\ble-entry2 & ! consequ5ce ( x is1 t ! sum ( all ! debtor items 9 ! :ole book is equal to ! sum ( all ! cr$itor items4 ,= :at is ! fir/ set b ! second ) ! items 9 a diff]5t ord]8 ,if x 7 conv5i5t1 "o entry ( ea* sum might be made a d\ble-entry4 ,! multiplication table is call$ a table ( d\ble-entry1 because #db1 = in/ance1 ?\gh x occurs only once1 appe>s 9 two diff]5t aspects1 namely1 z #f times #g & z #g times #f4 ,suppose1 = example1 t !re are five acc\nts1 ,a1 ,b1 ,c1 ,d1 ,e1 & t ea* acc\nt has "o transaction ( its own ) e o!r acc\nt2 & let ! debits be 9 ! columns1 ! cr$its 9 ! r[s1 z foll[s3 ---------------------------------------- --- ,debtor ,a ,b ,c ,d ,e ---------------------------------------- --- ,a1 ,cr$itor #bc #ai #cb #d ------------------------------ ,b1 ,cr$itor #ag #f #aa #be ------------------------------ ,c1 ,cr$itor #i #da #aj #b ------------------------------ ,d1 ,cr$itor #ad #bh #af #c ------------------------------ ,e1 ,cr$itor #ae #d #fj #a ------------------------------ ,"h ! #af is suppos$ to appe> 9 ,d's acc\nt z ,d cr$itor by ,c1 & 9 ,c's acc\nt z ,c debtor to ,d4 ,& to say t ! sum ( debtor items is ! same z t ( cr$itor items1 is m]ely to say t ! prec$+ numb]s give ! same sum1 :e!r ! r[s or ! columns be fir/ add$ up4 ,if x be desir$ to close ! l$g] :5 x /&s z above1 ! foll[+ is ! way ! acc\nts w /&3 ! l9es 9 italics w pres5tly be expla9$4 ,a1 ,debtor4 ,a1 ,cr$itor4 ,b1 ,debtor4 ,b1 ,cr$itor4 ,to ,b #ag ,by ,b #bc ,to ,a #bc ,by ,a #ag ,to ,c #i ,by ,c #ai ,to ,c #da ,by ,c #f ,to ,d #ad ,by ,d #cb ,to ,d #bh ,by ,d #aa ,to ,e #ae ,by ,e #d ,to ,e #d ,by ,e #be ,to ,balance #bc ,by ,balance #cg -- -- -- -- #gh #gh #if #if ---------------------------------------- ---------------------- ,c1 ,debtor4 ,c1 ,cr$itor4 ,d1 ,debtor4 ,d1 ,cr$itor4 ,to ,a #ai ,by ,a #i ,to ,a #cb ,by ,a #ad ,to ,b #f ,by ,b #da ,to ,b #aa ,by ,b #bh ,to ,d #af ,by ,d #aj ,to ,c #aj ,by ,c #af ,to ,e #fj ,by ,e #b ,to ,e #a ,by ,e #c ,by ,balance #ci ,to ,balance #g --- --- -- -- #aja #aja #fa #fa ---------------------------------------- ------------------------ ,e1 ,debtor4 ,e1 ,cr$itor4 ,balance1 ,debtor4 ,balance1 ,cr$4 ,to ,a #d ,by ,a #ae ,to ,b #cg ,by ,a #bc ,to ,b #be ,by ,b #d ,to ,c #ci ,by ,d #g ,to ,c #b ,by ,c #fj ,by ,e #df -- -- ,to ,d #c ,by ,d #a #gf #gf ,to ,balance #df -- -- #hj #hj ,9 all ! "p ( ! above :i* is pr9t$ 9 ,roman lett]s we see no?+ b ! prec$+ table repeat$4 ,b :5 all ! acc\nts h be5 complet$1 & no m entries are left to be made1 !re rema9s ! la/ process1 :i* is t]m$ balanc+ ! l$g]4 ,to get an idea ( ?is1 suppose a new cl]k1 :o goes r\nd all ! acc\nts1 collect+ debts & cr$its1 & tak+ !m all ^u himself1 t he alone may be entitl$ to claim ! debts & to be responsible = ! assets ( ! conc]n4 ,to ?is new cl]k1 :om ,i w call ! balance-cl]k1 e acc\nt gives up :at x has1 :e!r ! same be debt or cr$it4 ,! ca%-cl]k gives up all ! ca%2 ! cl]ks ( ! two k9ds ( bills give up all _! docum5ts1 :e!r bills receivable or entries ( bills payable 7rememb] t any entry aga9/ :i* !re is money set d[n 9 ! books c\nts z money :5 giv5 up1 t is1 z money due or money ow+72 ! cl]ks ( ! sev]al acc\nts ( goods give up all _! unsold rema9d]s at co/ prices2 ! cl]ks ( ! sev]al p]sonal acc\nts give up v\*]s = ! sums ow+ to or f ! sev]al p>ties2 & s on4 ,b :]e m has be5 paid out ?an receiv$1 ! balance-cl]k adju/s ^! acc\nts by giv+ in/ead ( receiv+2 9 fact1 he s acts z to make ! debtor & cr$itor sides ( ! acc\nts he visits equal 9 am\nt4 ,= in/ance1 ! ,a acc\nt is indebt$ to ! conc]n #ee1 :ile paym5ts or dis*>ges to ! am\nt ( #gh h be5 made by x4 ,! balance-cl]k accord+ly h&s ov] #bc to t acc\nt1 = :i* x becomes debtor1 :ile ! balance ent]s itself z cr$itor to ! same am\nt4 ,b 9 ! ,b acc\nt !re is #if ( receipt1 & only #ei ( paym5t or dis*>ge4 ,! balance-cl]k !n receives #cg f ?is acc\nt1 :i* is !re=e cr$it$ by balance1 :ile ! balance ackn[l$ges z mu* ( debt4 ,! balance acc\nt mu/1 ( c\rse1 exactly balance itself1 if ! acc\nts be all "r2 = ( all ! equal & opposite entries ( :i* ! l$g] consi/s1 s f> z !y d n balance "o ano!r1 "o goes into "o side ( ! balance acc\nt1 & ! o!r into ! o!r4 ,?us ! balance acc\nt becomes a te/ ( ! accuracy ( "o "p ( ! "w3 if its two sides d n give ! same sums1 ei!r !re h be5 entries :i* h n _h _! correspond+ balanc+ entries correctly made1 or else !re has be5 error 9 ! additions4 ,b s9ce ! balance acc\nt mu/ ?us always give ! same sum on bo? sides1 & s9ce balance debtor implies :at is fav\rable to ! conc]n1 & balance cr$itor :at is unfav\rable1 does x n appe> z if ?is sy/em c\ld only be appli$ to cases 9 :i* !re is nei!r loss nor ga98 ,?is br+s u to ! two acc\nts 9 :i* are ent]$ all t ! conc]n began )1 & all t x ga9s or loses--! /ock acc\nt1 & ! pr(it-&-loss acc\nt4 ,9 ord] to make all t !re 0 to beg9 ) a matt] ( d\ble entry1 ! op5+ ( ! l$g] supposes ! m]*ant himself to put 8 sev]al cl]ks 9 *>ge ( _! sev]al dep>tm5ts4 ,9 ! /ock acc\nt1 /ock1 :i* "h /&s = ! own] ( ! books1 is made cr$itor by all ! prop]ty1 & debtor by all ! liabilities2 :ile ! sev]al acc\nts are made debtors = all !y take f ! /ock1 & cr$itors by all ! responsibilities !y und]take4 ,suppose1 = in/ance1 !re are #ejj 9 ca% at ! comm5cem5t ( ! l$g]4 ,!re w !n appe> t ! m]*ant has h&$ ov] to ! ca%-box #ejj1 & 9 ! /ock acc\nt w appe>1 8,/ock cr$itor by ca%1 #ejj20 :ile 9 ! ca% acc\nt w appe>1 8,ca% debtor to /ock1 #ejj40 ,suppose t at ! beg9n+ !re is a debt out/&+ ( #ej to ,smi? & ,co41 !n !re w appe> 9 ! /ock acc\nt1 8,/ock debtor to ,smi? & ,co4 #ej10 & 9 ,smi? & ,co4's acc\nt w appe> 8,smi? & ,co4 cr$itors by /ock1 #ej40 ,?us !re is d\ble entry = all t ! conc]n beg9s ) by ?is contrivance ( ! /ock acc\nt4 ,! acc\nt to :i* ev]y?+ is plac$ = :i* an actual equival5t is n se5 9 ! books is ! pr(it-&-loss acc\nt4 ,?is pr(it-&-loss acc\nt1 or ! cl]k :o keeps x1 is made answ]able = e loss1 & ! suppos$ cause ( e ga94 ,?is acc\nt1 !n1 becomes debtor = e loss1 & cr$itor by e ga94 ,if goods be damag$ to ! am\nt ( #bj by accid5t1 & a loss to t am\nt occur 9 _! sale1 say !y co/ #hj & sell = #fj ca%1 x is cle> t !re is an entry 8,ca% debtor to goods #fj10 & 8,goods cr$itor by ca% #fj40 ,n[1 !re is an entry ( #hj some:]e to ! debit ( ! goods = ca% laid out1 or bills giv51 = ! :ole ( ! goods4 ,x w\ld affect ! accuracy ( ! acc\nts to take no notice ( ?is2 = :5 ! balance-cl]k comes to adju/ ?is acc\nt1 he w\ld f9d he receives #bj less ?an he might h reckon$ ^u1 )\t any explanation ( ! reason2 & !re w\ld be a failure ( ! pr9ciple ( d\ble-entry4 ,s9ce x is conv5i5t t ! balance acc\nt ( ! goods %\ld m]ely repres5t ! /ock 9 h& at ! close1 ! acc\nt ( goods !re=e lays ! responsibility ( #bj ^u ! pr(it-&-loss acc\nt1 or !re is ! entry 8,goods cr$itor by pr(it-&-loss1 #bj10 & also 8,pr(it-&-loss debtor to goods1 #bj40 ,aga91 9 all paym5ts :i* are n to br+ 9 a specific return1 su* z h\se & trade exp5ses1 wages1 c4 ^! sev]al acc\nts are suppos$ to adju/ matt]s ) ! pr(it-&-loss acc\nt be=e ! balance beg9s4 ,?us1 suppose ! outgo+s f ! m]e premises occupi$ exce$ any?+ ^? premises yield by #bjj1 or ! debits ( ! h\se acc\nt exce$ its cr$its by #bjj1 ! acc\nt %\ld be balanc$ by transf]r+ ! responsibility to ! pr(it-&-loss acc\nt1 "u ! entries 8,h\se exp5ses cr$itor by pr(it-&-loss1 #bjj01 8,pr(it-&-loss debtor to h\se exp5ses1 #bjj40 ,9 ?is way ! pr(it-&-loss acc\nt /eps 9 f "t to "t be=e ! balance acc\nt comm5ces its op]ations1 9 ord] t t same balance acc\nt may consi/ ( no?+ b ! necess>y matt]s ( acc\nt = ! next ye>'s l$g]4 ,?is transf]5ce ( acc\nts1 or transfusion ( "o acc\nt into ano!r1 requires att5tive consid]ation4 ,! receiv+ acc\nt becomes cr$itor = ! cr$its1 & debtor = ! debits1 ( ! transmitt+ acc\nt4 ,! rule1 !re=e1 is3 ,make ! transmitt+ acc\nt balance itself1 &1 on :i*ev] side x is necess>y to ent] a balanc+ sum1 make ! acc\nt debtor or cr$itor1 z ! case may be1 to ! receiv+ acc\nt1 & ! latt] cr$itor or debtor to ! =m]4 ,?us1 suppose acc\nt ,a is to be transf]r$ to acc\nt ,b1 & ! latt] is to arrange ) ! balance acc\nt4 ,if ! two /& z 9 ,roman lett]s1 ! processes 9 ,italic lett]s w occur be=e ! f9al close4 ,a1 ,debtor4 ,a1 ,cr$itor4 ,b1 ,debtor4 ,b1 ,cr$itor4 ,to sundries #ajj;,by sundries #ejj,to sundries #fjj;,by sundries #djj ,to ,b 4 4 4 #djj ,to ,balance #bjj;,by ,a 4 4 4 #djj ---- ---- ---- ---- #ejj #ejj #hjj #hjj ,& ! entry 9 ! balance acc\nt w be1 8,cr$itor by ,b1 #bjj10 %ew+ t1 on ^! two acc\nts1 ! cr$its exce$ ! debits by #bjj4 ,/ill1 be=e ! balance acc\nt is made up1 x is desirable t ! pr(it-&-loss acc\nt %\ld be transf]r$ to ! /ock acc\nt2 = ! pr(it & loss ( ?is ye> is ( no mom5t z a "p ( next ye>'s l$g]1 except 9 s f> z x affects ! /ock at ! comm5cem5t ( ! latt]4 ,let ?is be done1 & ! balance acc\nt may !n be made 9 ! =m requir$4 ,! /ock acc\nt & ! pr(it-&-loss acc\nt1 ! latt] be+ ! only direct *annel ( alt]ation = ! =m]1 diff] 9 a peculi> mann]#fj f ! o!r prelim9>y acc\nts1 & ! balance acc\nt is a species ( umpire4 ,!y repres5t ! m]*ant3 _! int]e/s are 8 int]e/s2 he is solv5t ^u ! excess ( _! cr$its ov] _! debits1 insolv5t ^u ! excess ( _! debits ov] _! cr$its4 ,x is exactly ! rev]se 9 all ! o!r acc\nts4 ,if a malici\s p]son 7 to get at ! l$g]1 & put on a ciph] to ! p\nds 9 v>i\s items1 ) a view ( mak+ ! conc]n appe> worse ?an x really is1 he w\ld make 8 alt]ations on ! debtor sides ( ! /ock & pr(it-&-loss acc\nts1 & on ! cr$itor sides ( all ! o!rs4 ,accord+ly1 9 ! balance acc\nt1 ! net /ock1 aft] ! incorporation ( ! pr(it-&-loss acc\nt1 appe>s on ! cr$itor side 7if n1 x %\ld be call$ am\nt ( insolv5cy1 n /ock71 & ! debts ( ! conc]n appe> on ! same side4 ,b on ! debit side ( ! balance acc\nt appe> all ! assets ( ! conc]n 7= :i* ! balance-cl]k is debtor to ! cl]ks f :om he has tak5 !m74 #fj ,! treatises on book-keep+ h describ$ ?is diff]5ce 9 z peculi> a mann]4 ,!y call ^! acc\nts ! fictiti\s acc\nts4 ,n[ !y repres5t ! m]*ant himself2 _! cr$its are ga9 to ! bus9ess1 _! debits losses or liabilities4 ,if ! t]ms real & fictiti\s are to be us$ at all1 !y are ! real acc\nts1 end all ! o!rs are z fictiti\s z ! cl]ks :om we h suppos$ to keep !m4 ,! "y /ud5t mu/ endeav\r to get ! enl>g$ view ( ! words debtor & cr$itor :i* is requisite1 & mu/ !n le>n by practice 7= no?+ else w give x7 facility 9 allott+ ! actual entries 9 ! wa/e-book to ! prop] sides ( ! prop] acc\nts4 ,i d n "h pret5d to give m ?an su* a view ( ! subject z may assi/ him 9 /udy+ a treatise on book-keep+1 :i* he w probably f9d to conta9 little m ?an examples4 ,,app5dix ,,viii4 ,,on ,,! ,,r$uction ,,( ,,fractions ,,to ,,o!rs ,,( ,,ne>ly ,,equal ,,value4 ,!re is a useful me?od ( f9d+ fractions :i* %all be ne>ly equal to a giv5 fraction1 & ) :i* ! comput] ought to be acqua9t$4 ,proce$ z 9 ! rule = f9d+ ! greate/ common measure ( ! num]ator & d5om9ator1 & br+ all ! quoti5ts into a l9e4 ,!n write d[n1 #a #bnd ,quot4 -------- ------------------------ #a/ ,quot4 #a/ ,quot4 #b;d ,quot4 #a ,!n take ! ?ird quoti5t1 multiply ! num]ator & d5om9ator ( ! second by x1 & add to ! products ! prec$+ num]ator & d5om9ator4 ,=m a ?ird fraction ) ! results = a num]ator & d5om9ator4 ,!n take ! f\r? quoti5t1 & proce$ ) ! ?ird & second fractions 9 ! same way2 & s on till ! quoti5ts are exhau/$4 ,= example1 let ! fraction be _/4 #iaca7acabh7a1 #b #aacg #ciig7c1 #a #eea #ehf7a1 #ae #bja #ce7a1 #b #bf #i7a1 #h #h #a ,?is is ! process = f9d+ ! greate/ common measure ( #iaca & #acabh 9 its mo/ compact =m1 & ! quoti5ts & fractions are3 #a #b #c #a #a #ae #a #b #a #h #a #b #g #i #af #bdi #bfe #ggi #ajdd #iaca --- --- --- --- ---- ----- ----- ----- ------ ------ #a #c #aj #ac #bc #ceh #cha #aabj #aeja #acabh ,x w be se5 t we h ?us a set ( fractions end+ ) ! orig9al fraction itself1 & =m$ by ! above rule1 z foll[s3 #a #a #a/ ,fraction -------- --- #a/ ,quot4 #a #b;d ,quot4 #b #b;d ,fraction ------------------------ --- #a/ ,quot4 #b;d ,quot4 #a #c #b;d ,num4 #c;d ,quot4 #a/ ,num4 #b #c #a #g #c;d ,fraction ------------------------------- --------- --- #b;d ,d54 #c;d ,quot4 #a/ ,d54 #c #c #a #aj #c;d ,num4 #d? ,quot4 #b;d ,num4 #g #a #b #i #d? ,fraction ------------------------------- --------- ---2 #c;d ,d54 #d? ,quot4 #b;d ,d54 #aj #a #c #ac & s on4 ,b we h done some?+ m ?an m]ely reasc5d to ! orig9al fraction by means ( ! quoti5ts4 ,! set ( fractions1 _/1 _/1 _/1 _/1 c4 are cont9ually approa*+ 9 value to ! orig9al fraction1 ! fir/ be+ too great1 ! second too small1 ! ?ird too great1 & s on alt]nately1 b ea* "o be+ ne>] to ! giv5 fraction ?an any ( ^? be=e x4 ,?us1 _/ is too great1 & _/ is too small2 b _/ is n s mu* too small z _/ is too great4 ,& aga91 _/1 ?\gh too great1 is n s mu* too great z _/ is too small4 ,moreov]1 ! diff]5ce ( any ( ! fractions f ! orig9al fraction is nev] great] ?an a fraction hav+ unity = its num]ator & ! product ( ! d5om9ator & ! next d5om9ator = its d5om9ator4 ,?us1 _/ does n err by s mu* z _/1 nor _/ by s mu* z _/1 nor _/ by s mu* z _/1 nor _/ by s mu* z _/1 c4 ,la/ly1 no fraction ( a less num]ator & d5om9ator c come s ne> to ! giv5 fraction z any "o ( ! fractions 9 ! li/4 ,?us1 no fraction ) a less num]ator ?an #bdi1 & a less d5om9ator ?an #ceh1 c come s ne> to #iaca #bdi ----- z ---4 #acabh #ceh ,! read] may take any example = himself1 & ! te/ ( ! accuracy ( ! process is ! ultimate return to ! fraction begun )4 ,ano!r te/ is z foll[s3 ,! num]ator ( ! diff]5ce ( any two consecutive approximat+ fractions ought to be unity4 ,?us1 9 our in/ance1 we h _/ & _/1 :i*1 ) a common d5om9ator1 #bc #ceh1 h #egbh & #egbg = _! num]ators4 ,z ano!r example1 let u exam9e ?is "q3 ,! l5g? ( ! ye> is #cfebdbbd days1 :i* is call$ 9 common life #cfe days4 ,take ! fraction _/1 & proce$ z 9 ! rule4 #bdbbd7ajjjjj7d1 #g1 #a1 #d1 #i1 #b #bdif #cajd #fd #fjh #j #cb #a #g #h #ci #cei #geg --- --- --- ---- ---- ---- #d #bi #cc #afa #adhb #cabe & _/ is #bdbbd 9 its l[e/ t]ms4 ,h5ce1 x appe>s t ! excess ( ! ye> ov] #cfe days am\nts to ab\t #a day 9 #d ye>s1 :i* is n wrong by s mu* z #a day 9 #aaf ye>s2 m accurately1 to #g days 9 #bi ye>s1 :i* is n wrong by s mu* z #a day 9 #ieg ye>s2 m accurately /ill1 to #h days 9 #cc ye>s1 :i* is n wrong by s mu* z #a day 9 #ecac ye>s2 & s on4 ,?is me?od may be appli$ to f9d+ fractions ne>ly equal to ! squ>e roots ( integ]s1 9 ! foll[+ mann]3 #dc #f 444 #f #a #e #d #e #e #d #e #a #f #f #a #e #d1 c4 #a #g #f #c #i #b #i #c #f #g #a #g #f #c1 c4 ------------------------------ #f #a #a #c #a #e #a #c #a #a #a #ba #a #c1 c4 ,set d[n ! numb] ^: squ>e root is want$1 say #dc4 ,?is squ>e root is #f & a fraction4 ,set d[n ! integ] #f 9 ! fir/ & ?ird r[1 & #a 9 ! second r[ always4 ,=m ! successive r[s ea* f ! "o be=e1 9 ! foll[+ mann]3 ,"o r[ ,! next r[ has b1 a1 c1 =m$ 9 ?is ord]1 be+ ?us1 a a excess ( bc1 already =m$1 ov] a4 b b quoti5t ( #dc - a divid$ by b4 c c integ] 9 ! quoti5t ( #f a divid$ by b4 ,?us ! second r[ is =m$ f ! fir/1 z "u3 #fa excess ( #g #a 7bo? j f\nd7 ov] #f4 #ag #dc - #f #f divid$ by #a4 ---- #fa integ] ( #f #f divid$ by #g 7j f\nd74 ,! ?ird r[ is =m$ f ! second1 ?us3 #a #e excess ( #a #f ov] #a4 #g #f #dc - #a #a divid$ by #g4 #a #a integ] ( #f #a divid$ by #f2 & s on4 ,9 process ( "t ! second column1 #a1 #g1 #a1 occurs aga91 aft] :i* ! sev]al columns are repeat$ 9 ! same ord]4 ,z a f9al process1 take ! set 9 ! l[e/ l9e 7exclud+ ! fir/1 #f71 namely1 #a1 #a1 #c1 #a1 #e1 #a1 #c1 c4 & use !m by ! rule giv5 at ! beg9n+ ( ?is article1 z foll[s3 #a #a #c #a #e #a #c #a #a1 c4 #a #a #d #e #bi #cd #aca #afe #bif --- --- --- --- ---- ---- ---- ---- ---- #a #b #g #i #eb #fa #bce #bif #eca ,h5ce1 #f_/ is v ne> ! squ>e root ( #dc1 n err+ by s mu* z #a ---------4 #bif #eca ,if we try x1 we %all f9d 7_/7 to be _/1 ! squ>e ( :i* is _/1 or #dc_/4 ,?is rule is ( use :5 x is frequ5tly want$ to use "o squ>e root1 & !re=e desirable to asc]ta9 :e!r any easy approximation exi/s by means ( a common fraction4 ,= example1 #b is (t5 us$4 #b #a 444 #aa #a #aa #a #ab #b #b #b #b #b #a #b #e #ab #bi #gj --- --- --- --- --- ---1 c4 #b #e #ab #bi #gj #afi ,"h x appe>s t #bi #a #ii #ajj - #a #a---- does n err by --------2 consequ5tly1 ---- or ------- is1 #gj #gj #afi #gj #gj consid]+ ! ease ( ! op]ation1 a fair approximation4 ,9 fact1 _/ is #adadbheg 444 ! tru? be+ #adadbace 444 ,! foll[+ is an additional example3 #ai #d 444 #d #b #c #c #b #d #d #b #a #c #e #b #e #c #a #c #d #b #a #c #a #b #h #b #a #c #a #b1 c4 #a #a #d #e #ad --- --- --- --- ---1 c4 #b #c #aa #ad #ci ,,app5dix ,,ix4 ,,on ,,"s ,,g5]al ,,prop]ties ,,( ,,numb]s4 ,,prop4 #a4 ,if a fraction be r$uc$ to its l[e/ t]ms1 s call$1#fa t is1 if nei!r1 num]ator nor d5om9ator be divisible by any integ] great] ?an unity1 !n no fraction ( a small] num]ator & d5om9ator c h ! same value4 #fa ,?is !orem %ews t :at is call$ r$uc+ a fraction to its l[e/ t]ms 7namely1 divid+ num]ator & d5om9ator by _! greate/ common measure71 is correctly s call$4 ,let a_/b be a fraction 9 :i* a & b h no common measure great] ?an unity3 &1 if possible1 let c_/d be a fraction ( ! same value1 c be+ less ?an a1 & d less ?an b4 ,n[1 s9ce a c a b --- ---1 we h --- ---2 b d c d let m be ! integ] quoti5t ( ^! la/ fractions 7:i* mu/ exi/1 s9ce a c1 b d71 & let e & f be ! rema9d]s4 ,!n a mc e c mc --- or ---------- --- ---- b md f d md ,h5ce1 e mc --- & ---- mu/ be equal1 = if n1 f md mc e mc e ---------- w\ld lie betwe5 ---- & ---1 md f md f in/ead ( be+ equal to ! =m]4 ,h5ce1 a e --- ---2 b f s t if a fraction ^: num]ator & d5om9ator h no common measure great] ?an unity1 be equal to a fraction ( l[] num]ator & d5om9ator1 x is equal to ano!r 9 :i* ! num]ator & d5om9ator are /ill l[]4 ,if we proce$ ) a e --- --- 9 a simil> mann]1 we f9d b f a g --- --- :]e g e1 h f1 b h & s on4 ,n[1 if !re be any process :i* p]petually dim9i%es ! t]ms ( a fraction by "o or m units at e /ep1 x mu/ at la/ br+ ei!r ! num]ator or d5om9ator1 or bo?1 to #j4 ,let a v --- --- b w be "o ( ! /eps1 & let a kv x1 b kw y2 s t kv x v ---------- ---4 kw y w ,n[1 if x #j b n y1 ?is is absurd1 = x gives kv kv ---------- ----4 kw y kw ,a simil> absurdity foll[s if y be #j1 b n x2 & if bo? x & y be #j1 !n a kv1 b kw1 or a & b h a common measure1 k4 ,n[ k mu/ be great] ?an #a1 = v & w are less ?an c & d1 :i* by hypo!sis are less ?an a & b4 ,consequ5tly a & b h a common measure k great] ?an #a1 :i* by hypo!sis !y h n4 ,if1 !n1 a & b be integ]s n divisible by any integ] great] ?an #a1 ! fraction a_/b is really 9 its l[e/ t]ms4 ,also a & b are said to be prime to "o ano!r4 ,,prop4 #b4 ,if ! product ab be divisible by c1 & if c be prime to b1 x mu/ divide a4 ,let ab b d ---- d1 !n --- ---4 c c c ,n[ b_/c is 9 its l[e/ t]ms2 !re=e1 by ! la/ proposition1 d & a mu/ h a common measure4 ,let ! greate/ common measure be k1 & let a kl1 d km4 ,!n b km m m --- ---- ---1 & --- c kl l l is also 9 its l[e/ t]ms2 b s is b_/c2 !re=e we mu/ h m b1 l c1 = o!rwise a fraction 9 its l[e/ t]ms w\ld be equal to ano!r ( l[] t]ms4 ,!re=e a kc1 or a is divisible by c4 ,& f ?is x foll[s1 t if a numb] be prime to two o!rs1 x is prime to _! product4 ,let a be prime to b & c1 !n no measure ( a c measure ei!r b or c1 & no su* measure c measure ! product bc2 = any measure ( bc :i* is prime to "o mu/ measure ! o!r4 ,,prop4 #c4 ,if a be prime to b1 x is prime to all ! p[]s ( b4 ,e measure#fb ( a is prime to b1 & !re=e does n divide b4 ,h5ce1 by ! la/1 no measure ( a divides b2 h5ce1 a is prime to b1 & s is e measure ( x2 !re=e1 no measure ( a divides bb1 consequ5tly a is prime to b1 & s on4 ,h5ce1 if a be prime to b1 a _c divide )\t rema9d] any p[] ( b4 ,?is is ! reason :y no fraction c be made into a decimal unless its d5om9ator be measur$ by no prime#fc numb]s except #b & #e4 ,= if a c --- ---1 b #aj :i* la/ is ! g5]al =m ( a decimal fraction1 let a #aj;a --- be 9 its l[e/ t]ms2 !n ------ b b is an integ]1 :5ce 7,prop4 #b7 b mu/ divide #aj1 & s mu/ all ! divisors ( b4 ,if1 !n1 among ! divisors ( b !re be any prime numb]s except #b & #e1 we h a prime numb] 7:i* is ( c\rse a numb] prime to #aj7 n divid+ #aj1 b divid+ "o ( its p[]s1 :i* is absurd4 #fb ,= t :i* measures a measure is itself a measure2 s t if a measure ( a c\ld h a measure 9 common ) b1 a itself w\ld h a common measure ) b4 #fc ,a prime numb] is "o :i* is prime to all numb]s except its own multiples1 or has no divisors except #a & itself4 ,,prop4 #d4 ,if b be prime to a1 all ! multiples ( b1 z b1 #b;b1 444 up to 7a-#a7;b mu/ leave diff]5t rema9d]s :5 divid$ by a4 ,= if1 m be+ great] ?an n1 & bo? less ?an a1 we h mb & nb giv+ ! same rema9d]1 x foll[s t mb-nb1 or 7m-n7b1 is divisible by a2 :5ce 7,prop4 #b71 a divides m-n1 a numb] less ?an itself1 :i* is absurd4 99 99 99 99 99 ,if a numb] be divid$ into its prime factors1 or r$uc$ to a product ( prime numb]s only 7z 9 #cfj #b #b #b #c #c #e71 & if a1 b1 c1 c4 be ! prime factors1 & 1 1 1 c4 ! numb] ( times !y sev]ally ent]1 s t ! numb] is a b c c41 !n ?is c be done 9 only "o way3 ,= any prime numb] v1 n includ$ 9 ! above li/1 is prime to a1 & !re=e to a1 to b & !re=e to b & !re=e to a b ,proce$+ 9 ?is way1 we prove t v is prime to ! complete product above1 or to ! giv5 numb] itself4 ,! numb] ( divisors :i* ! prec$+ numb] abc 444 c h1 #j & itself includ$1 is 7 #a77 #a77 #a74444 ,= a z ! divisors #a1 a1 a 444 a & no o!rs1 #a 9 all4 ,simil>ly1 b has #a divisors1 & s on4 ,n[ z all ! divisors are made by multiply+ toge!r "o out ( ea* set1 _! numb] 7page #bjb7 is 7 #a77 #a77 #a74444 ,if a numb]1 n1 be divisible by c]ta9 prime numb]s1 say #c1 #e1 #g1 #aa1 !n ! ?ird "p ( all ! numb]s up to n is divisible by #c1 ! fif? "p by #e1 & s on4 ,b m ?an ?is3 :5 ! multiples ( #c are omitt$1 exactly ! fif? "p ( ^? :i* rema9 are divisible by #e2 = ! fif? "p ( ! :ole are divisible by #e1 & ! fif? "p ( ^? :i* are remov$ are divisible by #e1 !re=e ! fif? "p ( ^? :i* are left are divisible by #e4 ,aga91 because ! sev5? "p ( ! :ole are divisible by #g1 & ! sev5? "p ( ^? :i* are divisible by #c1 or by #e1 or by #ae1 x foll[s t :5 all ^? :i* are multiples ( #c or #e1 or bo?1 are remov$1 ! sev5? "p ( ^? :i* rema9 are divisible by #g2 & s on4 ,h5ce1 ! numb] ( numb]s n exce$+ n1 :i* are n divisible by #c1 #e1 #g1 or #aa1 is _/ ( _/ ( _/ ( _/ ( n4 ,proce$+ 9 ?is way1 we f9d t ! numb] ( numb]s :i* are prime to n1 t is1 :i* are n divisible by any "o ( its prime factors1 a1 b1 c1 444 is a - #a b - #a c - #a n ------- ------- ------- 444 a b c or a-#a - #a;bc 444 7a - #a77;b - #a77;c - #a74444 ,?us1 #cfj be+ #bce1 its numb] ( divisors is #d #c #b1 or #bd1 & !re are #bc4a4b4d or #if numb]s less ?an #cfj :i* are prime to x4 ,,prop4 #e4 ,if a be prime to b1 !n ! t]ms ( ! s]ies1 a1 a1 a1 444 sev]ally divid$ by b1 mu/ all leave diff]5t rema9d]s1 until #a occurs z a rema9d]1 aft] :i* ! cycle ( rema9d]s w be aga9 repeat$4 ,let a b give ! rema9d] r 7n unity72 !n a b gives ! same rema9d] z ra b1 :i* 7,prop4 #d7 _c be r3 let x be s4 ,!n a b gives ! same rema9d] z sa b1 :i* 7,prop4 #d7 _c be ei!r r or s1 unless s be #a3 let x be t4 ,!n a b gives ! same rema9d] z ta b2 if t be n #a1 ?is _c be ei!r r1 s1 or t3 let x be u4 ,s we g on gett+ diff]5t rema9d]s1 until #a occurs z a rema9d]2 aft] :i*1 at ! next /ep1 ! rema9d] ( a b is repeat$4 ,n[1 #a mu/ come at la/2 = division by b _c give any rema9d]s b #j1 #a1 #b1 444 b- #a2 & #j nev] arrives 7,prop4 #c71 s t z soon z b-#b diff]5t rema9d]s h occurr$1 no "o ( :i* is unity1 ! next1 :i* mu/ be diff]5t f all t prec$e1 mu/ be #a4 ,if n be=e1 !n at a we mu/ h a rema9d] #a2 aft] :i* ! cycle w obvi\sly be repeat$4 ,?us1 #g1 #g1 #g1 #g1 c4 w1 :5 divid$ by #e1 be f\nd to give ! rema9d]s #b1 #d1 #c1 #a1 c4 ,,prop4 #f4 ,! diff]5ce ( two m? p[]s is always divisible )\t rema9d] by ! diff]5ce ( ! roots2 or a -b is divisible by a-b2 = a - b a - ab ab - b a7a - b7 b7a - b7 ,f :i*1 if a-b is divisible by a -b1 s is a-b4 ,b a-b is divisible by a-b2 s !re=e is a- b2 s !re=e is a-b2 & s on4 ,!re=e1 if a & b1 divid$ by c1 leave ! same rema9d]1 a & b1 a & b1 c4 sev]ally divid$ by c1 leave ! same rema9d]s2 = ?is means t a-b is divisible by c4 ,b a - b is divisible by a-b1 & !re=e by e measure ( a-b1 or by c2 b a-b _c be divisible by c1 unless a & b1 sev]ally divid$ by c1 give ! same rema9d]4 ,,prop4 #g4 ,if b be a prime numb]1 & a be n divisible by b1 !n a & 7a-#a7 #a leave ! same rema9d] :5 divid$ by b4 ,?is proposition _c be prov$ "h1 z x requires a little m ( algebra ?an ! read] ( ?is "w possesses4#fd #fd ,exp& 7a-#a7 by ! b9omial !orem2 %ew t :5 b is a prime numb] e coeffici5t :i* is n unity is divisible by b2 & ! proposition foll[s4 ,,prop4 #h4 ,9 ! la/ case1 a divid$ by b leaves a rema9d] #a4 ,f ! la/1 a-a leaves ! same rema9d] z 7a-#a7 #a-;a or 7a-#a7- 7a-#a72 t is1 ! rema9d] ( a-a is n alt]$ if a be r$uc$ by a unit4 ,by ! same rule1 x may be r$uc$ ano!r unit1 & s on1 /ill )\t any alt]ation ( ! rema9d]4 ,at la/ x becomes #a-a1 or #j1 ! rema9d] ( :i* is #j4 ,accord+ly1 a-a1 :i* is a7a- #a71 is divisible by b2 & s9ce b is prime to a1 x mu/ 7,prop4 #b7 divide a-#a2 t is1 a1 divid$ by b1 leaves a rema9d] #a1 if b be a prime numb] & a be n divisible by b4 ,f ! above x appe>s 7,prop4 #e & #g71 t if a be prime to b1 ! set #a1 a1 a1 a1 c4 successively divid$ by b1 give a set ( rema9d]s beg9n+ ) #a1 & 9 :i* #a occurs aga9 at a1 if n be=e1 & at a c]ta9ly 7:e!r be=e or n71 if b be a prime numb]4 ,f ! po9t at :i* #a occurs1 ! cycle ( rema9d]s recomm5ces1 & #a is always ! beg9n+ ( a cycle4 ,if1 !n1 a be ! fir/ p[] :i* gives #a = rema9d]1 m mu/ ei!r be b-#a1 or a measure ( x1 :5 b is a prime numb]4 ,b if we divide ! t]ms ( ! s]ies m1 ma1 ma1 ma1 c4 by b1 m be+ less ?an b1 we h cycles ( rema9d]s beg9n+ ) m4 ,if #a1 r1 s1 t1 c4 be ! fir/ set ( rema9d]s1 !n ! second set is ! set ( rema9d]s aris+ f m1 mr1 ms1 mt1 c4 ,if #a nev] occur 9 ! fir/ set be=e a 7except at ! beg9n+71 !n all ! numb]s "u b-#a inclusive are f\nd among ! set #a1 r1 s1 t1 c42 & if m be prime to b 7,prop4 #d71 all ! same numb]s are f\nd1 9 a diff]5t ord]1 among ! rema9d]s ( m1 mr1 c4 ,b %\ld x happ5 t ! set #a1 r1 s1 t1 c4 is n complete1 !n m1 mr1 ms1 c4 may give a diff]5t set ( rema9d]s4 ,all ^! la/ !orems are con/antly v]ifi$ 9 ! process = r$uc+ a fraction to a decimal fraction4 ,if m be prime to b1 or ! fraction m_/b 9 its l[e/ t]ms1 ! process involves ! successive division ( m1 m #aj1 m #aj1 c4 by b4 ,?is process c nev] come to an end unless "s p[] ( #aj1 say #aj1 is divisible by b2 :i* _c be1 if b conta9 any prime factors except #b & #e4 ,9 e o!r case ! quoti5t repeats itself1 ! repeat+ "p sometimes comm5c+ f ! fir/ figure1 sometimes f a lat] figure4 ,?us1 _/ yields #adbhegadbheg1 c41 b _/ gives #jg7adbheg77adbheg71 c41 & _/ gives #jc7egadbh77egadbh71 c4 ,9 m_/b1 ! quoti5t always repeats f ! v beg9n+ :5ev] b is a prime numb] & m is less ?an b2 & ! numb] ( figures 9 ! repeat+ "p is !n always b-#a1 or a measure ( x4 ,t x mu/ be s1 appe>s f ! above propositions4 ,be=e proce$+ f>!r1 we write d[n ! repeat+ "p ( a quoti5t1 ) ! rema9d]s :i* are left aft] ! sev]al figures are =m$4 ,let ! fraction be _/1 we h #jehhbcebidaagfdg ,?is may be read ?us3 #aj by #ag1 quoti5t #j1 rema9d] #aj2 #aj by #ag1 quoti5t #je1 rema9d] #ae2 #aj by #ag1 quoti5t #jeh1 rema9d] #ad2 & s on4 ,x ?us appe>s t #aj by #ag leaves a rema9d] #a1 :i* is accord+ to ! !orem4 ,if we multiply #jehh1 c4 by any numb] "u #ag1 ! same cycle is obta9$ ) a diff]5t beg9n+4 ,?us1 if we multiply by #ac1 we h #gfdgjehhbcebidaa beg9n+ ) :at comes aft] rema9d] #ac 9 ! fir/ numb]4 ,if we multiply by #g1 we h #daag1 c4 ,! reason is obvi\s3 _/ #ac1 or _/1 :5 turn$ into a decimal fraction1 />ts ) ! divisor #acj1 & we proce$ j z we d 9 =m+ _/1 :5 )9 f\r figures ( ! close ( ! cycle4 ,x w also be se51 t 9 ! la/ half ( ! cycle ! quoti5t figures are complem5ts to #i ( ^? 9 ! fir/ half1 & t ! rema9d]s are complem5ts to #ag4 ,?us1 9 #jehh1 c4 & #idaa1 c4 we see #j #i #i1 #e #d #i1 #h #a #i1 c41 & #aj #g #ag1 #ae #b #ag1 #ad #c #ag1 c4 ,we may %ew ! necessity ( ?is z foll[s3 ,if ! rema9d] #a nev] occur till we come to use a1 !n1 b be+ prime1 b-#a is ev52 let x be #bk4 ,accord+ly1 a-#a is divisible by b2 b ?is is ! product ( a-#a & a #a1 "o ( :i* mu/ be divisible by b4 ,x _c be a-#a1 = !n a p[] ( a prec$+ ! 7b-#a7? w\ld leave rema9d] #a1 :i* is n ! case 9 our in/ance3 x mu/ !n be a #a1 s t a divid$ by b leaves a rema9d] b-#a2 & ! k? /ep concludes ! fir/ half ( ! process4 ,accord+ly1 9 our in/ance1 we see1 b be+ #ag & a be+ #aj1 t rema9d] #af occurs at ! #h? /ep ( ! process4 ,at ! next /ep1 ! rema9d] is t yield$ by #aj7;b-#a71 or #i;b b-#aj1 :i* gives ! rema9d] b-#aj4 ,b ! fir/ rema9d] ( all 0 #aj1 & #aj 7b-#aj7 b4 ,if ev] ?is complem5tal "* occur 9 any /ep1 x mu/ cont9ue1 :i* we %ew z foll[s3 ,let r be a rema9d]1 & b-r a subsequ5t rema9d]1 ! sum be+ b4 ,at ! next /ep aft] ! fir/ rema9d]1 we divide #ajr by b1 &1 at ! next /ep aft] ! second rema9d]1 we divide #aj;b-#ajr by b4 ,n[1 s9ce ! sum ( #ajr & #aj;b-#ajr is divisible by b1 ! two rema9d]s f ^! new /eps mu/ be su* z add$ toge!r w give b1 & s on2 & ! quoti5ts add$ toge!r mu/ give #i1 = ! sum ( ! rema9d]s #ajr & #aj;b-#ajr yields a quoti5t #aj1 ( :i* ! two rema9d]s give #a4 ,if _/ & _/ be tak51 ! repeat+ p>ts w be f\nd to conta9 #eh & #fj figures4 ,( ^! we write d[n only ! fir/ halves1 z ! read] may supply ! re/ by ! complem5tal prop]ty j giv54 #jafidiaebedbcgbhhaceeicbbjcch1 c4 #jafcicddbfbbiejhaifgbacaadgedj1 c4 ,"h1 !n1 are two numb]s1 ! fir/ ( :i* multipli$ by any numb] "u #ei1 & ! second by any numb] "u #fa1 c h ! products =m$ by c>ry+ c]ta9 ( ! figures f "o end to ! o!r4 ,b1 b be+ /ill prime1 x may happ5 t rema9d] #a may occur be=e b-#a figures are obta9$2 9 :i* case1 z %ewn1 ! numb] ( figures mu/ be a measure ( b-#a4 ,= example1 take _/4 ,! repeat+ quoti5t1 writt5 z above1 has only #e figures1 & #e measures #da-a4 #jbdci ,n[1 ?is p]iod1 x w be f\nd1 has its figures m]ely transpos$1 if we multiply by #aj1 #ah1 #af1 or #cg4 ,b if we multiply by any o!r numb] "u #da1 we conv]t ?is p]iod into ! p]iod ( ano!r fraction ^: d5om9ator is #da4 ,! foll[+ are #h p]iods :i* may be f\nd4 #jbdci #abaie #jdhgh #adfcd #jgcag #bfhbi #jigef #cfehe ,to f9d m_/#da1 look out = m among ! rema9d]s1 & take ! p]iod 9 :i* x is1 beg9n+ aft] ! rema9d]4 ,?us1 _/ is #hbibfhbibf1 c41 & _/ is #cfehecfehe1 c4 ,^! p]iods are complem5tal1 f\r & f\r1 z #jbdci & #igefj1 #jgcag & #ibfhb1 c4 ,& if ! fir/ numb]1 #jbdci1 be multipli$ by any numb] "u #da1 look = t numb] among ! rema9d]s1 & ! product is f\nd 9 ! p]iod ( t rema9d] by beg9n+ aft] ! rema9d]4 ,?us1 #jbdci multipli$ by #bc gives #efjig1 & by #f gives #adfcd4 ,! read] may try to deciph] = himself h[ x is t1 ) no m figures ?an ! foll[+1 we c ext5d ! result ( our division4 ,! fraction ( :i* ! p]iod is to be f\nd is _/4 #hg7ajj7jaadidbe #acj #dcj #hbj #jaadidbe #be #cgj #bhgcefbe #be #bbj #gahcijfbe #be #dfj #agieigfefbe #be #be #ddhiidadjfbe #jaadidbebhgcefbe #gahcijfbe #agieigf #efbe #ddhiid ---------------------------------- #jaadidbebhgcefcbahcijhjdeiggjaadid ,,app5dix ,x4 ,,on ,,comb9ations4 ,!re are "s ?+s connect$ ) comb9ations :i* ,i place 9 an app5dix1 because ,i int5d to demon/rate !m m briefly ?an ! matt]s 9 ! text4 ,suppose a numb] ( boxes1 say #d1 9 ea* ( :i* !re are c\nt]s1 say #e1 #g1 #c1 & #aa sev]ally4 ,9 h[ _m ways c "o c\nt] be tak5 out ( ea* box1 ! ord] ( go+ to ! boxes n be+ reg>d$4 ,answ]1 9 #e #g #c #aa ways4 ,= out ( ! fir/ box we may draw a c\nt] 9 #e diff]5t ways1 & to ea* su* draw+ we may annex a draw+ f ! second 9 #g diff]5t ways--giv+ #e #g ways ( mak+ a draw+ f ! fir/ two4 ,to ea* ( ^! we may annex a draw+ f ! ?ird box 9 #c ways--giv+ #e #g #c draw+s f ! fir/ ?ree2 & s on4 ,! foll[+ /atem5ts may n[ be easily demon/rat$1 & simil> ones made z to o!r cases4 ,if ! ord] ( go+ to ! boxes make a diff]5ce1 & if a1 b1 c1 d be ! numb]s ( c\nt]s 9 ! sev]al boxes1 !re are #d #b #c #a a b c d di/9ct ways4 ,if we want to draw1 say #b out ( ! fir/ box1 #c out ( ! second1 #a out ( ! ?ird1 & #c out ( ! f\r?1 & if ! ord] ( ! boxes be n consid]$1 ! numb] ( ways is a - #a b - #a b - #b d - #a d - #b a------- b------- -------- c d------- ------- #b #b #c #b #c ,if ! ord] ( go+ to ! boxes be consid]$1 we mu/ multiply ! prec$+ by #d #c #b #a4 ,if ! ord] ( ! draw+s out ( ! boxes makes a diff]5ce1 b n ! ord] ( ! boxes1 !n ! numb] ( ways is a7a-#a7;b7b-#a77;b-#b7;cd7d-#a77;d-#b7 ,! n? p[] ( a1 or a1 repres5ts ! numb] ( ways 9 :i* a c\nt]s diff]5tly m>k$ c be di/ribut$ 9 n boxes1 ord] ( plac+ !m 9 ea* box n be+ consid]$4 ,suppose we want to di/ribute #d diff]5tly-m>k$ c\nt]s among #g boxes4 ,! fir/ c\nt] may g into ei!r box1 :i* gives #g ways2 ! second c\nt] may g into ei!r2 & any ( ! fir/ #g allotm5ts may be comb9$ ) any "o ( ! second #g1 giv+ #g #g di/9ct ways2 ! ?ird c\nt] v>ies ea* ( ^! 9 #g diff]5t ways1 giv+ #g #g #g 9 all2 & s on4 ,b if ! c\nt]s be undi/+ui%able1 ! problem is a v diff]5t ?+4 ,requir$ ! numb] ( ways 9 :i* a numb] c be comp\nd$ ( o!r numb]s1 diff]5t ord]s c\nt+ z diff]5t ways4 ,?us1 #a #c #a & #a #a #c are to be consid]$ z di/9ct ways ( mak+ #e4 ,x w be obvi\s1 on a little exam9ation1 t ea* numb] c be compos$ 9 exactly twice z _m ways z ! prec$+ numb]4 ,take #h = in/ance4 ,if e possible way ( mak+ #g be writt5 d[n1 #h may be made ei!r by increas+ ! la/ compon5t by a unit1 or by annex+ a unit at ! end4 ,?us1 #a #c #b #a may yield #a #c #b #b1 or #a #c #b #a #a3 & all ! ways ( mak+ #h w ?us be obta9$2 = any way ( mak+ #h1 say a b c d1 mu/ proce$ f ! foll[+ mode ( mak+ #g1 a b c 7d-#a74 ,n[1 7d-#a7 is ei!r #j--t is1 d is unity & is /ruck out--or 7d-#a7 rema9s1 a numb] #a less ?an d4 ,h5ce x foll[s t ! numb] ( ways ( mak+ n is #b4 ,= !re is obvi\sly #a way ( mak+ #a1 #b ( mak+ #b2 !n !re mu/ be1 by our rule1 #b ways ( mak+ #c1 #b ways ( mak+ #d2 & s on4 #a #a #a #a #a #a #a #a #a #a #a #b #a #b #a #b #a #a #a #c #b #a #b #a #a #b #b #b #c #c #a #d ,?is table exhibits ! ways ( mak+ #a1 #b1 #c1 & #d4 ,h5ce x foll[s 7:i* ,i leave ! read] to inve/igate7 t !re are twice z _m ways ( =m+ a b z !re are ( =m+ a & !n annex+ to x a =mation ( b2 f\r times z _m ways ( =m+ a b c z !re are ( annex+ to a =mation ( a =mations ( b & ( c2 & s on4 ,also1 9 summ+ numb]s :i* make up a b1 !re are ways 9 :i* a is a re/1 & ways 9 :i* x is n1 & z _m ( "o z ( ! o!r4 ,requir$ ! numb] ( ways 9 :i* a numb] c be comp\nd$ ( odd numb]s1 diff]5t ord]s c\nt+ z diff]5t ways4 ,if a be ! numb] ( ways 9 :i* n c be s made1 & b ! numb] ( ways 9 :i* n #a c be made1 !n a b mu/ be ! numb] ( ways 9 :i* n #b c be made2 = e way ( mak+ #ab out ( odd numb]s is ei!r a way ( mak+ #aj ) ! la/ numb] increas$ by #b1 or a way ( mak+ #aa ) a #a annex$4 ,?us1 #a #e #c #c gives #ab1 =m$ f #a #e #c #a giv+ #aj4 ,b #a #i #a #a is =m$ f #a #i #a giv+ #aa4 ,consequ5tly1 ! numb] ( ways ( =m+ #ab is ! sum ( ! numb] ( ways ( =m+ #aj & ( =m+ #aa4 ,n[1 #a c only be =m$ 9 #a way1 & #b c only be =m$ 9 #a way2 h5ce #c c only be =m$ 9 #a #a or #b ways1 #d 9 only #a #b or #c ways4 ,if we take ! s]ies #a1 #a1 #b1 #c1 #e1 #h1 #ac1 #ba1 #cd1 #ee1 #hi1 c4 9 :i* ea* numb] is ! sum ( ! two prec$+1 !n ! n? numb] ( ?is set is ! numb] ( ways 7ord]s c\nt+7 9 :i* n c be =m$ ( odd numb]s4 ,?us1 #aj c be =m$ 9 #ee ways1 #aa 9 #hi ways1 c4 ,%ew t ! numb] ( ways 9 :i* mk c be made ( numb]s divisible by m 7ord]s c\nt+7 is #b4 ,9 ! two s]ies1 #a #a #a #b #c #d #f #i #ac #ai #bh1 c4 #j #a #j #a #a #a #b #b #c #d #e1 c41 ! fir/ has ea* new t]m aft] ! ?ird equal to ! sum ( ! la/ & la/ b two2 ! second has ea* new t]m aft] ! ?ird equal to ! sum ( ! la/ b "o & la/ b two4 ,%ew t ! n? numb] 9 ! fir/ is ! numb] ( ways 9 :i* n c be made up ( numb]s :i*1 divid$ by #c1 leave a rema9d] #a2 & t ! n? numb] 9 ! second is ! numb] ( ways 9 :i* n c be made up ( numb]s :i*1 divid$ by #c1 leave a rema9d] #b4 ,x is v easy to %ew 9 h[ _m ways a numb] c be made up ( a giv5 numb] ( numb]s1 if diff]5t ord]s c\nt z diff]5t ways4 ,suppose1 = in/ance1 we w\ld "k 9 h[ _m ways #ab c be ?us made ( #g numb]s4 ,if we write d[n #ab units1 !re are #aa int]vals betwe5 unit & unit4 ,!re is no way ( mak+ #ab out ( #g numb]s :i* does n answ] to di/ribut+ #f p>tition-m>ks 9 ! int]vals1 #a 9 ea* ( #f1 & collect+ all ! units :i* are n sep>at$ by p>tition-m>ks4 ,?us1 #a #a #c #b #a #b #b1 :i* is "o way ( mak+ #ab out ( #g numb]s1 answ]s to #a #a #aaa #aa #a #aa #aa 9 :i* ! p>tition-m>ks come 9 ! #a/1 #b;d1 #e?1 #g?1 #h?1 & #aj? ( ! #aa int]vals4 ,consequ5tly1 to ask 9 h[ _m ways #ab c be made ( #g numb]s1 is to ask 9 h[ _m ways #f p>tition-m>ks c be plac$ 9 #aa int]vals2 or1 h[ _m comb9ations or selections c be made ( #f out ( #aa4 ,! answ] is1 #aa #aj #i #h #g #f -----------------------1 or #dfb4 #a #b #c #d #e #f ,let u d5ote by m ! numb] ( ways 9 :i* m ?+s c be tak5 out ( n ?+s1 s t m is ! abbreviation = n - #a n - #b n - m #a n ------ ------- 444 z f> z ------------- #b #c m ,!n m also repres5ts ! numb] ( ways 9 :i* m #a numb]s c be put toge!r to make n #a4 ,:at we prov$ above is1 t #f is ! numb] ( ways 9 :i* we c put toge!r #g numb]s to make #ab4 ,!re w n[ be no difficulty 9 prov+ ! foll[+3 #b #a #a #b #c 444 n ,9 ! prec$+ "q1 #j did n ent] into ! li/ ( numb]s us$4 ,?us1 #c #a #j #j 0 n consid]$ z "o ( ! ways ( putt+ toge!r f\r numb]s to make #e4 ,b let u n[ ask1 :at is ! numb] ( ways ( putt+ toge!r #g numb]s to make #ab1 all[+ #j to be 9 ! li/ ( numb]s4 ,!re c be no m 7nor few]7 ways ( do+ ?is ?an ( putt+ #g numb]s toge!r1 among :i* #j is n includ$1 to make #ai4 ,take e way ( mak+ #ab 7#j includ$71 & put on #a to ea* numb]1 & we get a way ( mak+ #ai 7#j n includ$74 ,take any way ( mak+ #ai 7#j n includ$71 & /rike (f #a f ea* numb]1 & we h "o ( ! ways ( mak+ #ab 7#j includ$74 ,accord+ly1 #f is ! numb] ( ways ( putt+ toge!r #g numb]s 7#j be+ all[$7 to make #ab4 ,& 7m- #a7 is ! numb] ( ways ( putt+ toge!r m numb]s to make n1 #j be+ includ$4 ,?is la/ am\nts to ! solution ( ! foll[+3 ,9 h[ _m ways c n c\nt]s 7undi/+ui%able f ea* o!r7 be di/ribut$ into m boxes8 ,& ! foll[+ w n[ be easily prov$3 ,! numb] ( ways ( di/ribut+ c undi/+ui%able c\nt]s into b boxes is 7b - #a7;b c - #a1 if any box or boxes may be left empty4 ,b if !re mu/ be #a at lea/ 9 ea* box1 ! numb] ( ways is 7b - #a7;c - #a2 if !re mu/ be #b at lea/ 9 ea* box1 x is 7b - #a7;c- b-#a2 if !re mu/ be #c at lea/ 9 ea* box1 x is 7b - #a7;c - #b;b - #a2 & s on4 ,! numb] ( ways 9 :i* m odd numb]s c be put toge!r to make n1 is ! same z ! numb] ( ways 9 :i* m ev5 numb]s 7#j includ$7 c be put toge!r to make n-m2 & ?is is ! numb] ( ways 9 :i* m numb]s 7odd or ev51 #j includ$7 c be put toge!r to make 7n-m74 ,accord+ly1 ! numb] ( ways 9 :i* m odd numb]s c be put toge!r to make n is ! same z ! numb] ( comb9ations ( m-#a ?+s out ( 7n-m7 m-#a1 or 7n m7-#a4 ,unless n & m be bo? ev5 or bo? odd1 ! problem is evid5tly impossible4 ,!re are curi\s & useful relations exi/+ betwe5 numb]s ( comb9ations1 "s ( :i* may readily be exhibit$1 "u ! simple expression ( m to /& = ! numb] ( ways 9 :i* m ?+s may be tak5 out ( n4 ,suppose we h to take #e out ( #ab3 ,let ! #ab ?+s be m>k$ ,a1 ,b1 ,c1 c4 & set ap>t "o ( !m1 ,a4 ,e collection ( #e out ( ! #ab ei!r does or does n include ,a4 ,! numb] ( ! latt] sort mu/ be #e2 ! numb] ( ! =m] sort mu/ be #d1 s9ce x is ! numb] ( ways 9 :i* ! o!r f\r c be *os5 out ( all b ,a4 ,consequ5tly1 #e mu/ be #e #d1 & ?us we prove 9 e case1 m m 7m - #a7 #j & n bo? are #a2 = !re is b "o way ( tak+ none1 & b "o way ( tak+ all4 ,& aga9 m & 7n-m7 are ! same ?+s4 ,& if m be great] ?an n1 m is #j2 = !re are no ways ( do+ x4 ,we make "o ( our prec$+ results m symmetrical if we write x ?us1 #b #j #a #b 444 n ,if we n[ write d[n ! table ( symbols 9 :i* ! 7m #a7? #j #a #b #c1 c4 -------------------------------------- #a #j #a #b #c1 c4 #b #j #a #b #c1 c4 #c #j #a #b #c1 c4 c4 c4 c4 c4 c4 numb] ( ! n? r[ repres5ts m1 ! numb] ( comb9ations ( m out ( n1 we see x prov$ above t ! law ( =mation ( ?is table is z foll[s3 ,ea* numb] is to be ! sum ( ! numb] above x & ! numb] prec$+ ! numb] above x4 ,n[1 ! fir/ r[ mu/ be #a1 #a1 #j1 #j1 #j1 c4 & ! fir/ column mu/ be #a1 #a1 #a1 #a1 c4 s t we h a table ( ! foll[+ k9d1 :i* may be c>ri$ z f> z we please3 #j #a #b #c #d #e #f #g #h #i #aj ---------------------------------------- ------------ #a #a #a #j #j #j #j #j #j #j #j #j #b #a #b #a #j #j #j #j #j #j #j #j #c #a #c #c #a #j #j #j #j #j #j #j #d #a #d #f #d #a #j #j #j #j #j #j #e #a #e #aj #aj #e #a #j #j #j #j #j #f #a #f #ae #bj #ae #f #a #j #j #j #j #g #a #g #ba #ce #ce #ba #g #a #j #j #j #h #a #h #bh #ef #gj #ef #bh #h #a #j #j #i #a #i #cf #hd #abf #abf #hd #cf #i #a #j #aj #a #aj #de #abj #baj #beb #baj #abj #de #aj #a ,?us1 9 ! r[ #i1 "u ! column head$ #d1 we see #abf1 :i* is #i #h #g #f 7#a #b #c #d71 ! numb] ( ways 9 :i* #d c be *os5 out ( #i1 :i* we repres5t by #d-i4 ,if we add ! sev]al r[s1 we h #a #a or #b1 #a #b #a or #b1 next #a #c #c #a or #b1 c4 :i* v]ify a !orem already ann\nc$2 & ! law ( =mation %ews u t ! sev]al columns are =m$ ?us3 #a #a #a #b #a #a #c #c #a #a #a #a #b #a #a #c #c #a ----- ------- --------- #a #b #a #a #c #c #a #a #d #f #d #a1 c4 s t ! sum 9 ea* r[ mu/ be d\ble ( ! sum 9 ! prec$+4 ,b we c c>ry ! consequ5ces ( ?is mode ( =mation fur!r4 ,if we make ! p[]s ( #a x by actual algebraical multiplication1 we see t ! process makes ! same oblique addition 9 ! =mation ( ! num]ical multipli]s ( ! p[]s ( x4 #a x #a x ------- #a x x x --------------- #a #bx x #a #bx x #a x --------------- #a #bx x x #bx x ----------------------- #a #cx #cx x ,"h are ! second & ?ird p[]s ( #a x3 ! f\r?1 we c tell be=eh& f ! table1 mu/ be #a #dx #fx #dx x2 & s on4 ,h5ce we h 7#a x7 #j #ax #bx #cx 444 nx :i* is usually writt5 ) ! symbols #j1 #a1 c4 at l5g?1 ?us1 n - #a n - #a n - #b 7#a x7 #a nx n-------x n------- -------x c4 #b #b #c ,?is is ! simple/ case ( :at 9 algebra is call$ ! b9omial !orem4 ,if in/ead ( #a x we use x a1 we get 7xa7 x#a;ax#b;ax#c;ax444 na ,we c make ! same table 9 ano!r =m4 ,if we take a r[ ( ciph]s beg9n+ ) unity1 & sett+ d[n ! fir/1 add ! next1 & !n ! next1 & s on1 & !n repeat ! process ) "o /ep less1 & !n aga9 ) "o /ep less1 we h ! foll[+3 #a #j #j #j #j #j #j #a #a #a #a #a #a #a #a #b #c #d #e #f #a #c #f #aj #ae #a #d #aj #bj #a #e #ae #a #f #a ,9 ! oblique columns we see #a #a1 #a #b #a1 #a #c #c #a1 c4 ! same z 9 ! orig9al table1 & =m$ by ! same additions4 ,if1 be=e mak+ ! additions1 we _h always multipli$ by a1 we %\ld h got ! sev]al compon5ts ( ! p[]s ( #a a1 ?us1 #a #j #j #j #j #a a a a a #a #b;a #c;a #d;a #a #c;a #f;a #a #d;a #a :]e ! oblique columns #a a1 #a #b;a a1 #a #c;a #c;a a1 c41 give ! sev]al p[]s ( #a a4 ,if in/ead ( beg9n+ ) #a1 #j1 #j1 c4 we _h begun ) p1 #j1 #j1 c4 we %\ld h got p1 p #d;a1 p #f;a1 c4 at ! bottom ( ! sev]al columns2 & if we _h writt5 at ! top x1 x1 x1 x1 #a1 we %\ld h _h all ! mat]ials = =m+ p7x a7 by multiply+ ! t]ms at ! top & bottom ( ea* column toge!r1 & add+ ! results4 ,suppose we foll[ ?is mode ( =m+ p7x a7 q7x a7 r7x a7 s4 x x x #a x x #a x #a #a p #j #j #j q #j #j r #j #c p pa pa pa q qa qa r ra p #bpa #cpa q #bqa r p #cpa q p px #cpax #cpax pa qx #bqax qa rx ra s px 7#cpa q7x 7#cpa #bqa r7x pa qa ra s ,n[1 obs]ve t all ?is might be done 9 "o process1 by ent]+ q1 r1 & s "u _! prop] p[]s ( x 9 ! fir/ process1 z foll[s x x x #a p q r s p pa q pa qa r pa qa ra s p #bpa q #cpa #bqa r p #cpa q p ,?is process#fe is ! "o us$ 9 ,app5dix ,,xi41 ) ! slight alt]ation ( v>y+ ! sign ( ! la/ lett]1 & mak+ subtractions in/ead ( additions 9 ! la/ column4 ,z x /&s1 x is ! mo/ conv5i5t mode ( writ+ x a in/ead ( x 9 a l>ge class ( algebraical expressions4 ,= in/ance1 :at does #bx x #cx #gx #i become :5 x #e is writt5 in/ead ( x8 ,! expression1 made complete1 is1 #bx #ax #jx #cx #gx #i #a #j #c #g #i #b #aa #ee #bgh #acig #fiid #b #ba #afj #ajgh #fghg #b #ca #cae #bfec #b #da #ebj #b #ea ,answ]1 #bx #eax #ebjx #bfecx #fghgx #fiid4 #fe ,! pr9ciple ( ?is mode ( demon/ration ( ,horn]'s me?od 0 /at$ 9 ,"y's ,algebra 7#ahbc71 be+ ! e>lie/ elem5t>y "w 9 :i* t me?od 0 giv54 ,,app5dix ,,xi4 ,,on ,,horn]',s ,,me?od ,,( ,,solv+ ,,equations4 ,! rule giv5 9 ?is *apt] is ins]t$ on acc\nt ( its excell5ce z an ex]cise 9 computation4 ,! examples *os5 w require b little use ( algebraical signs1 t !y may be und]/ood by ^? :o "k no m ( algebra ?an is conta9$ 9 ! pres5t "w4 ,to solve an equation su* z #bx x - #cx #dafgic1 or1 z x is usually writt51 #bx x - #cx - #dafgic #j1 we mu/ fir/ asc]ta9 by trial n only ! fir/ figure ( ! root1 b also ! d5om9ation ( x3 if x be a #b1 = in/ance1 we mu/ "k :e!r x be #b1 or #bj1 or #bjj1 c41 or #b1 or #jb1 or #jjb1 c4 ,?is mu/ be f\nd by trial2 & ! %orte/ way ( mak+ ! trial is z foll[s3 ,write ! expression 9 its complete =m4 ,9 ! prec$+ case ! =m is n complete1 & ! complete =m is #bx #jx #ax - #cx - #dafgic4 ,to f9d :at ?is is :5 x is any numb]1 = in/ance1 #cjjj1 ! be/ way is to take ! fir/ multipli] 7#b71 multiply x by #cjjj1 & take 9 ! next multipli] 7#j71 multiply ! result by #cjjj1 & take 9 ! next multipli] 7#a71 & s on to ! end1 z foll[s3 #b #cjjj #j #fjjj2 #fjjj #cjjj #a #ahjjjjja #ahjjjjja #cjjj - #c #edjjjjjbiig #edjjjjjbiig #cjjj - #dafgic #afbjjjjjhegdbjg ,n[ try ! value ( ! above :5 x #cj4 ,we h !n1 = ! /eps1 #fj 7#b #cj #j71 #ahja1 #edjbg1 & la/ly1 #afbjhaj-dafgic1 or x #cj makes ! fir/ t]ms great] ?an #dafgic4 ,n[ try x #bj :i* gives #dj1 #hja1 #afjag1 & la/ly1 #cbjcdj-dafgic1 or x #bj makes ! fir/ t]ms less ?an #dafgic4 ,betwe5 #bj & #cj1 !n1 mu/ be a value ( x :i* makes #bx x-#cx equal to #dafgic4 ,& ?is is ! prelim9>y /ep ( ! process4 ,hav+ got ?us f>1 write d[n ! coeffici5ts #b1 #j1 #a1-c1 & -#dafgic1 ea* ) its prop] algebraical sign1 except ! la/1 9 :i* let ! sign be *ang$4 ,?is is ! mo/ conv5i5t way :5 ! la/ sign is-4 ,b if ! la/ sign be 1 x may be m conv5i5t to let x /&1 & *ange all :i* come be=e4 ,?us1 9 solv+ x-#abx #a #j1 we might write -#a #j #ab #a :]eas 9 ! in/ance be=e u1 we write #b #j #a -#c #dafgic ,hav+ done ?is1 take ! highe/ figure ( ! root1 prop]ly nam$1 :i* is #b t5s1 or #bj4 ,beg9 ) ! fir/ column1 multiply by #bj1 & jo9 x to ! numb] 9 ! next column2 multiply t by #bj1 & jo9 x to ! numb] 9 ! next column2 & s on4 ,b :5 y come to ! la/ column1 subtract ! product :i* comes out ( ! prec$+ column1 or jo9 x to ! la/ column aft] *ang+ its sign4 ,:5 ?is has be5 done1 repeat ! process ) ! numb]s :i* n[ /& 9 ! columns1 omitt+ ! la/1 t is1 ! subtract+ /ep2 !n repeat x aga91 go+ only z f> z ! la/ column b two1 & s on1 until ! columns pres5t a set ( r[s ( ! foll[+ appe>ance3 a b c d e f g h i k l m n o p to ! =mation ( :i* ! foll[+ is ! key3 f #bj;a b1 g #bj;f c1 h #bj;g d1 i e - #bj;h1 k #bj;a f1 l #bjk g1 m #bjl h1 n #bj;a k1 o #bjn l1 p #bj;a n4 ,we call ?is ,horn]'s ,process1 f ! "n ( its inv5tor4 ,! result is z foll[s3 #b #j #a -#c #dafgic 7#bj #dj #hja #afjag #ifdec #hj #bdja #fdjcg #abj #dhja #afj ,we h n[ be=e u ! r[ #b #afj #dhja #fdjcg #ifdec :i* furni%es our means ( guess+ at ! next1 or units' figure ( ! root4 ,call ! la/ column ! divid5d1 ! la/ b "o ! divisor1 & all t come be=e antec$5ts4 ,see h[ (t5 ! divid5d conta9s ! divisor2 ?is gives ! guess at ! next figure4 ,! guess is a true "o1#ff if1 on apply+ ,horn]'s process1 ! divisor result1 augm5t$ z x is by ! antec$5t processes1 /ill g z _m times 9 ! divid5d4 ,= example1 9 ! case be=e u1 #ifdec conta9s #fdjcg once2 let #a be put on its trial4 ,horn]'s process is f\nd to succe$1 & we h = ! second process1 #b #afj #dhja #fdjcg #ifdec #afb #difc #fijjj #bgdec #afd #eabg #gdabg #aff #ebic #afh ,z soon z we come to ! fractional portion ( ! root1 ! process assumes a m#fg me?odical =m4 ,! equation be+ ( ! f\r? degree1 annex f\r ciph]s to ! divid5d1 ?ree to ! divisor1 two to ! antec$5t1 & "o to ! previ\s antec$5t1 leav+ ! fir/ column z x is2 !n f9d ! new figure by ! divid5d & divisor1 z be=e1#fh & apply ,horn]'s process4 ,annex ciph]s to ! results1 z be=e1 & proce$ 9 ! same way4 ,! annex+ ( ! ciph]s prev5ts our hav+ any ?+ to d ) decimal po9ts1 & enables u to use ! quoti5t-figures )\t pay+ any att5tion to _! local values4 ,! foll[+ exhibits ! :ole process f ! beg9n+1 c>ri$ z f> z x is "h int5d$ to g be=e beg9n+ ! contraction1 :i* w give m figures1 z 9 ! rule = ! squ>e root4 ,! foll[+1 !n1 is ! process z f> z "o decimal place3 #ff ,v>i\s exceptions may arise :5 an equation has two ne>ly equal roots4 ,b ,i d n "h introduce algebraical difficulties2 & a /ud5t might give himself a hundr$ examples1 tak5 at haz>d1 )\t mu* *ance ( light+ ^u "o :i* gives any difficulty4 #fg ,?is =m might be also appli$ to ! integ] portions2 b x is h>dly ne$$ 9 su* in/ances z usually occur4 ,see ! article ,involution & ,evolution 9 ! ,supplem5t to ! ,p5ny ,cyclopdia4 #fh ,aft] ! second /ep1 ! trial w r>ely fail to give ! true figure4 #b #j #a -#c #dafgic7bac #dj #hja #afjag #ifdec #hj #bdja #fdjcg ----- #abj #dhja ----- #bgdecjjjj #afj ---- #fijjj #dgcciggh --- #difc #gdabgjjj --------- #afb #eabg -------- #afd #ebicjj #gegcjjgd #aff ------ #ggcdhcgf #afhj #ecdceh ---- #ecidcd #afhf #eddebh #afib #afih #agjd ---- ,if we n[ beg9 ! contraction1 x is good to "k be=eh& on :at numb] ( additional root-figures we may reckon4 ,we may be pretty c]ta9 ( hav+ ne>ly z _m z !re are figures 9 ! divisor :5 we beg9 to contract--"o less1 or at lea/ two less4 ,?us1 !re be+ n[ eight figures 9 ! divisor1 we may conclude t ! contraction w give u at lea/ six m figures4 ,to beg9 ! contraction1 let ! divid5d /&1 cut (f "o figure f ! divisor1 two f ! column be=e t1 ?ree f ! "o be=e t1 & s on4 ,?us1 our contraction beg9s ) #jjjb #agjd #eddebh #ggcdhcgf #dgcciggh ,! fir/ column is r5d]$ q useless "h4 ,conduct ! process z be=e1 us+ only ! figures :i* are n cut (f4 ,b x w be bett] to g z f> z ! fir/ figure cut (f1 c>ry+ f ! second figure cut (f4 ,we %all !n h z foll[s3 #agjd #eddebh #ggcdhcgf #dgcciggh7f #edeee #ggfgegjf #gcdced #edfeg #ghjjcfdh #edgei ,at ! next contraction ! column #agjd becomes #jjagjd1 & is q useless4 ,! next /ep1 sep>ately writt5 7:i* is n1 h[ev]1 necess>y 9 work+71 is #edgei #ghjjcfdh #gcdced7j ,"h ! divid5d #gcdced does n conta9 ! divisor #ghjjcf1 & we1 !re=e1 write #j z a root figure & make ano!r contraction1 or beg9 ) #edgei #ghjjcfdh #gcdced7i #ghjjhe #cbbgg #ghjacd ,at ! next contraction ! fir/ column becomes #jjedgei1 & is q useless1 s t ! rema9d] ( ! process is ! contract$ division4 #ghjacd7cbbgg7dacg #ajgb #bib #eh #c & ! root requir$ is #bacfjidacg4 ,i n[ write d[n ! complete process = ano!r equation1 "o root ( :i* lies betwe5 #c & #d3 x is x - #ajx #a #j #a #j -#aj -#a7caaajcijebjgcjiijgif #c -#a #bjjj #f #agjj #bjijjj #i #j #agia #aigfijjj #i #a #ahhcjj #gdccfijjjjjj #i #b #ahibca #agbcaagajbgcjjj #i #cj #aijafcjj #iiabdgddgfha #i #ca #aijbefca #cidfbhgedbj #i #cb #aijcdifcjj #j #j #aciadiaeei #i #cc #j #aijcebdbii #j #i #ehiicabc #i #cc #a #aijceebbih #b #g #j #j #ahhfjdg #i #cc #b #aijcefjfih #j #eia #agbhce #i #cc #cj #j #aijcefijig #h #efc #aeae #i #cc #cj #c #aijcefiadd #e #bb #ahc #i #cc #cj #f #aijcefiaia #ahh #ab #i #cc #cj #ij #aijcefiaic #jf #a #i #cccjii #aijcefiaidic #i #cccajh #jicccaag ,! /ud5t ne$ n repeat ! r[s ( figures s f> z !y come "u "o ano!r3 ?us1 x is n necess>y to repeat #aijcef4 ,b he mu/ use 8 own discretion z to h[ mu* x w\ld be safe = him to omit4 ,i h set d[n ! :ole process "h z a guide4 ,! foll[+ examples w s]ve = ex]cise3 #a4 #bx - #ajjx - #g #j x #gajehaacc4 #b4 x x x x #fjjj x #hecadcggbf4 #c4 x #cx - #dx - #aj #j x #ahiefidiafejd4 #d4 x #ajjx - #ex - #bagc #j x #dehbbdfjgajehdfd4 #e4 #b #abeiibajdihidhgcafdgfgbajfjgbgh4fi #f4 x - #fx #ajj x #ejgaceagdhgca4 #g4 x #bx #cx #cjj x #eieebeifgabbcih4 #h4 x x #ajjj x #iifffffgi4 #i4 #bgjjjx #bgjjjx #bfiiiiii x #iiffffff44444 #aj4 x - #fx #ajj x #ejgaceagdhg4 #aa4 x - #dx #gx - #hfc #j x #deaieejg4 #ab4 x - #bjx #h #j x #dffjjcgficjjjhgbgh4 #ac4 x x x - #aj #j x #agcgcgjbcc4 #ad4 x - #dfx - #cfx #ah #j x #dfgfafcjahdg1 or x #cdgafbcaib4 #ae4 x #dfx - #cfx - #ah #j x #aajhgibejcg4 #af4 #hiiax - #afbhchx #gdfbgax - #hajjj #j x #aaabbbcccdddeee4444 #ag4 #gbix - #dhfx #iix - #f #j x #aaaa4441 or #bbbb4441 or #cccc4444 #ah4 #bx #cx - #dx #ejj x #eicdhagifbcaeaebgi4 #ai4 x #bx x - #aej #j x #dffhdjijadeedaihcbecgdbiiabjagjehii4 #bj4 x x x #ejj x #hbdjifceehaddhehebfifc4 #ba4 x #bx #cx - #ajjjj #j x #bjhebijeebfjji4 #bb4 x - #dx - #bjjj #j x #dehadjjcfb4 #bc4 #ajx - #ccx - #aax - #ajj #j x #dadfgighjhehdbghghe4 #bd4 x x x x #abgfid x #ahfddhbcgcjie4 #be4 #ajx #aax #abx #ajjjjj x #baafeeiieeedejhhje4 #bf4 x x #ac x #bbjigeccjabjhhdi4 #bg4 x x - #dx - #afjj #j x #aadhbhcgaeg4 #bh4 x - #bx #e x #bjideeadhaedbcbfeiadhbchfedjegicjbifche gcjfajefbhbci4 #bi4 x - #hjx #bdx - #fx - #hjcgifci #j x #abc4gj #cj4 x - #bdbx - #fcaex #beggjif #j x #abc4ga #ca4 #bx - #cx #fx - #h #j x #adadbacefbcgcjiejdhhjc4gb #cb4 x - #aix #acbx - #cjbx #bjj #j x #ajbhjd1 or #d1 or #fegfec1 or #gciedcgc4 #cc4 #gx - #aax #fx #ex #bae x #bgjfdhjdichegia4gd #cd4 #gx #fx #ex #dx #cx #aa x #ggjgfhhaifbbfehebbcgibifeje4ge #ce4 #dx #gx #ix #fx #ex #cx #gib x #bjebjdbagfhgifjecfebadjdcdjabhabjaigcdf jbgeeiie #deedagbdbad4gf #cf4 #bahgx - #bdcjx #idex - #aejx #h #j x #aaaa44441 or #bbbb44441 or #cccc44441 or #dddd4444 #fi ,! solution ( x #jx #jx-#b #j4 #gj ,tak5 f a pap] on ! subject1 by ,mr4 ,pet] ,gray1 9 ! ,me*anics' ,magaz9e4 #ga ,tak5 f a pap] on ! subject1 by ,mr4 ,pet] ,gray1 9 ! ,me*anics' ,magaz9e4 #gb ,tak5 f a pap] on ! subject1 by ,mr4 ,pet] ,gray1 9 ! ,me*anics' ,magaz9e4 #gc ,tak5 f ! late ,mr4 ,pet] ,ni*olson's ,essay on ,involution & ,evolution4 #gd ,tak5 f ! late ,mr4 ,pet] ,ni*olson's ,essay on ,involution & ,evolution4 #ge ,tak5 f ! late ,mr4 ,pet] ,ni*olson's ,essay on ,involution & ,evolution4 #gf ,tak5 f ! late ,mr4 ,pet] ,ni*olson's ,essay on ,involution & ,evolution4 ,,app5dix ,,xii4 ,,rules ,,= ,,! ,,application ,,( ,,ari?metic ,,to ,,geometry4 ,! /ud5t %\ld make himself famili> ) ! mo/ common t]ms ( geometry1 aft] :i* ! foll[+ rules w pres5t no difficulty4 ,9 !m all1 x mu/ be und]/ood1 t :5 we talk ( multiply+ "o l9e by ano!r1 we mean ! repetition ( "o l9e z (t5 z !re are units ( a giv5 k9d1 z feet or in*es1 9 ano!r4 ,9 any o!r s5se1 x is absurd to talk ( multiply+ a quantity by ano!r quantity4 ,all quantities ( ! same k9d %\ld be repres5t$ 9 numb]s ( ! same unit2 ?us1 all ! l9es %\ld be ei!r feet & decimals ( a foot1 or in*es & decimals ( an in*1 c4 ,& 9 :atev] unit a l5g? is repres5t$1 a surface is express$ 9 ! correspond+ squ>e units1 & a solid 9 ! correspond+ cubic units4 ,?is be+ und]/ood1 ! rules apply to all sorts ( units4 ,to f9d ! area ( a rectangle4 ,multiply toge!r ! units 9 two sides :i* meet1 or multiply toge!r two sides :i* meet2 ! product is ! numb] ( squ>e units 9 ! area4 ,?us1 if #f feet & #e feet be ! sides1 ! area is #f #e1 or #cj squ>e feet4 ,simil>ly1 ! area ( a squ>e ( #f feet long is #f #f1 or #cf squ>e feet 7#bcd74 ,to f9d ! area ( a p>allelogram4 ,multiply "o side by ! p]p5dicul> di/ance betwe5 x & ! opposite side2 ! product is ! area requir$ 9 squ>e units4 ,to f9d ! area ( a trapezium4#gg ,multiply ei!r ( ! two sides :i* are n p>allel by ! p]p5dicul> let fall ^u x f ! middle po9t ( ! o!r4 #gg ,a f\r-sid$ figure1 :i* has two sides p>allel1 & two sides n p>allel4 ,to f9d ! area ( a triangle4 ,multiply any side by ! p]p5dicul> let fall ^u x f ! opposite v]tex1 & take half ! product4 ,or1 halve ! sum ( ! ?ree sides1 subtract ! ?ree sides sev]ally f ?is half sum1 multiply ! f\r results toge!r1 & f9d ! squ>e root ( ! product4 ,! result is ! numb] ( squ>e units 9 ! area2 & twice ?is1 divid$ by ei!r side1 is ! p]p5dicul> di/ance ( t side f its opposite v]tex4 ,to f9d ! radius ( ! int]nal circle :i* t\*es ! ?ree sides ( a triangle4 ,divide ! area1 f\nd 9 ! la/ p>agraph1 by half ! sum ( ! sides4 ,giv5 ! two sides ( a "r-angl$ triangle1 to f9d ! hypo!nuse4 ,add ! squ>es ( ! sides1 & extract ! squ>e root ( ! sum4 ,giv5 ! hypo!nuse & "o ( ! sides1 to f9d ! o!r side4 ,multiply ! sum ( ! giv5 l9es by _! diff]5ce1 & extract ! squ>e root ( ! product4 ,to f9d ! circumf]5ce ( a circle f its radius1 v ne>ly4 ,multiply twice ! radius1 or ! diamet]1 by #cadaeibg1 tak+ z _m decimal places z may be ?\ght necess>y4 ,= a r\gh computation1 multiply by #bb & divide by #g4 ,= a v exact computation1 9 :i* decimals %all be avoid$1 multiply by #cee & divide by #aac4 ,see 7#aca71 la/ example4 ,to f9d ! arc ( a circul> sector1 v ne>ly1 kn[+ ! radius & ! angle4 ,turn ! angle into seconds1#gh multiply by ! radius1 & divide ! product by #bjfbfe4 ,! result w be ! numb] ( units 9 ! arc4 #gh ,! "r angle is divid$ into #ij equal p>ts call$ degrees1 ea* degree into #fj equal p>ts call$ m9utes1 & ea* m9ute into #fj equal p>ts call$ seconds4 ,?us1 #b #ae #dj means #b degrees1 #ae m9utes1 & #dj seconds4 ,to f9d ! area ( a circle f its radius1 v ne>ly4 ,multiply ! squ>e ( ! radius by #cadaeibg4 ,to f9d ! area ( a sector1 v ne>ly1 kn[+ ! radius & ! angle4 ,turn ! angle into seconds1 multiply by ! squ>e ( ! radius1 & divide by #bjfbfe #b1 or #dabecj4 ,to f9d ! solid cont5t ( a rectangul> p>allelopip$4 ,multiply toge!r ?ree sides :i* meet3 ! result is ! numb] ( cubic units requir$4 ,if ! figure be n rectangul>1 multiply ! area ( "o ( its planes by ! p]p5dicul> di/ance betwe5 x & its opposite plane4 ,to f9d ! solid cont5t ( a pyramid4 ,multiply ! area ( ! base by ! p]p5dicul> let fall f ! v]tex ^u ! base1 & divide by #c4 ,to f9d ! solid cont5t ( a prism4 ,multiply ! area ( ! base by ! p]p5dicul> di/ance betwe5 ! opposite bases4 ,to f9d ! surface ( a sph]e4 ,multiply #d times ! squ>e ( ! radius by #cadaeibg4 ,to f9d ! solid cont5t ( a sph]e4 ,multiply ! cube ( ! radius by #cadaeibg _/1 or #dahhgi4 ,to f9d ! surface ( a "r cone4 ,take half ! product ( ! circumf]5ce ( ! base & slant+ side4 ,to f9d ! solid cont5t1 take "o-?ird ( ! product ( ! base & ! altitude4 ,to f9d ! surface ( a "r cyl9d]4 ,multiply ! circumf]5ce ( ! base by ! altitude4 ,to f9d ! solid cont5t1 multiply ! area ( ! base by ! altitude4 ,! weight ( a body may be f\nd1 :5 its solid cont5t is kn[n1 if ! weight ( "o cubic in* or foot ( ! body be kn[n4 ,b x is usual to =m tables1 n ( ! weights ( a cubic unit ( diff]5t bodies1 b ( ! proportion :i* ^! weights be> to "s "o among/ !m4 ,! "o *os5 is usually di/ill$ wat]1 & ! proportion j m5tion$ is call$ ! specific gravity4 ,?us1 ! specific gravity ( gold is #aicfb1 or a cubic foot ( gold is #aicfb times z heavy z a cubic foot ( di/ill$ wat]4 ,suppose n[ ! weight ( a sph]e ( gold is requir$1 ^: radius is #d in*es4 ,! cont5t ( ?is sph]e is #d #d #d #dahhh1 or #bfhjhcb cubic in*es2 & s9ce1 by 7#bag71 ea* cubic in* ( wat] weighs #bebdeh gra9s1 ea* cubic in* ( gold weighs #bebdeh #aicfb1 or #dhhhjia gra9s2 s t #bfhjhcb cubic in*es ( gold weigh #bfhjhcb #dhhhjia gra9s1 or #bbg p\nds troy ne>ly4 ,tables ( specific gravities may be f\nd 9 mo/ works ( *emi/ry & practical me*anics4 ,! cubic foot ( wat] is #ijhhdhh troy ounces1 #gegcgd troy p\nds1 #iigacfifia av]dupois ounces1 & #fbcbajfjf av]dupois p\nds4 ,= all r\gh purposes x w d to consid] ! cubic foot ( wat] z be+ #ajjj common ounces1 :i* r$uces tables ( specific gravities to common t]ms 9 an obvi\s way4 ,?us1 :5 we read ( a sub/ance :i* has ! specific gravity #daagb1 we may take x t a cubic foot ( ! sub/ance weighs #daag ounces4 ,= great] correctness1 dim9i% ?is result by #c p>ts out ( a ?\s&4 ,,! ,,end4 ,,walton ,,& ,,mab]ly',s ,,catalogue ,,( ,,educational ,,works1 ,,& ,,works ,,9 ,,sci5ce ,,& ,,g5]al ,,lit]ature4 ,,engli%4 ,dr4 ,r4 ,g4 ,la?am4 ,! ,engli% ,language4 ,f\r? ,edition4 #b vols4 #hvo4 #a #hs4 clo?4 ,la?am's ,elem5t>y ,engli% ,gramm>1 = ! ,use ( ,s*ools4 ,eighte5? ?\s&4 ,small #hvo4 #ds4 #f;d4 clo?4 ,la?am's ,h&-book ( ! ,engli% ,language1 = ! ,use ( ,/ud5ts ( ! ,univ]sities & high] ,classes ( ,s*ools4 ,?ird ,edition4 ,small #hvo4 #gs4 #f;d4 clo?4 ,la?am's ,logic 9 its ,application to ,language4 #abmo4 #fs4 clo?4 ,la?am's ,fir/ ,engli% ,gramm>1 adapt$ = g5]al use4 ,by ,,dr4 ,r4 ,g4 ,,la?am & an ,exp]i5c$ ,tea*]4 ,fcap4 #hvo4 ,la?am's ,hi/ory & ,etymology ( ! ,engli% ,language1 = ! ,use ( ,classical ,s*ools4 ,second ,edition1 revis$4 ,fcap4 #hvo4 #as4 #f;d4 cl4 ,mason's ,engli% ,gramm>1 includ+ ! ,pr9ciples ( ,grammatical ,analysis4 #abmo4 #cs4 #f;d4 clo?4 ,mason's ,c[p]'s ,task ,book ,i4 7! ,s(a71 ) ,notes on ! ,analysis & ,p>s+4 ,cr[n #hvo4 #as4 #f;d4 clo?4 ,abbott's ,fir/ ,engli% ,read]4 ,?ird ,edition4 #abmo41 ) ,illu/rations4 #as4 clo?1 limp4 ,abbott's ,second ,engli% ,read]4 ,?ird ,edition4 #abmo4 #as4 #f;d4 clo?1 limp4 ,,greek4 ,gre5wood's ,greek ,gramm>1 includ+ ,accid5ce1 ,irregul> ,v]bs1 & ,pr9ciples ( ,d]ivation & ,composition2 adapt$ to ! ,sy/em ( ,crude ,=ms4 ,small #hvo4 #es4 #f;d4 clo?4 ,khn]'s ,new ,greek ,delectus2 be+ ,s5t5ces = ,translation f ,greek into ,engli%1 & ,engli% into ,greek2 arrang$ 9 a sy/ematic ,progression4 ,translat$ & ,edit$ by ! late ,,dr4 ,,alex&] ,,all54 ,f\r? ,edition1 revis$4 #abmo4 #ds4 clo?4 ,gillespie's ,greek ,te/am5t ,roots1 9 a ,selection ( ,texts1 giv+ ! p[] ( ,read+ ! :ole ,greek ,te/am5t )\t difficulty4 ,) ,grammatical ,notes1 & a ,p>s+ ,lexicon associat+ ! ,greek ,primitives ) ,engli% ,d]ivatives4 ,po/ #hvo4 #gs4 #f;d4 clo?4 ,robson's ,con/ructive ,ex]cises = ,tea*+ ! ,elem5ts ( ! ,greek ,language1 on a sy/em ( ,analysis & ,syn!sis1 ) ,greek ,read+ ,lessons & copi\s ,vocabul>ies4 #abmo41 pp4 #djh4 #gs4 #f;d4 clo?4 ,robson's ,fir/ ,greek ,book4 ,ex]cises & ,read+ ,lessons ) ,copi\s ,vocabul>ies4 ,be+ ! ,fir/ ,"p ( ! 8,con/ructive ,greek ,ex]cises40 #abmo4 #cs4 #f;d4 clo?4 ,! ,london ,greek ,gramm>4 ,design$ to exhibit1 9 small ,compass1 ! ,elem5ts ( ! ,greek ,language4 ,six? ,edition4 #abmo4 #as4 #f;d4 clo?4 ,h>dy & ,adams's ,anabasis ( ,x5ophon4 ,expressly = ,s*ools4 ,) ,notes1 ,index ( ,names1 & a ,map4 #abmo4 #ds4 #f;d4 clo?4 ,smi?'s ,plato4 ,! ,apology ( ,socrates1 ,! ,crito1 & "p ( ! ,pha$o2 ) ,notes 9 ,engli% f ,/allbaum1 ,s*lei]ma*]'s ,introductions1 & 8 ,essay on ! ,wor? ( ,socrates z a ,philosoph]4 ,edit$ by ,dr4 ,,wm4 ,,smi?1 ,editor ( ! ,diction>y ( ,greek & ,roman ,antiquities1 c4 ,?ird ,edition4 #abmo4 #es4 clo?4 ,,lat94 ,new ,lat9 ,read+ ,book2 consi/+ ( ,%ort ,s5t5ces1 ,easy ,n>rations1 & ,descriptions1 select$ f ,caes>'s ,gallic ,w>2 9 ,sy/ematic ,progression4 ,) a ,diction>y4 ,second ,edition1 revis$4 #abmo4 #bs4 #f;d4 ,all5's ,new ,lat9 ,delectus2 be+ ,s5t5ces = ,translation f ,lat9 into ,engli%1 & ,engli% into ,lat92 arrang$ 9 a sy/ematic ,progression4 ,f\r? ,edition1 revis$4 #abmo4 #ds4 clo?4 ,! ,london ,lat9 ,gramm>2 includ+ ! ,eton ,syntax & ,prosody 9 ,engli%1 accompani$ ) ,notes4 ,sixte5? ,edition4 #abmo4 #as4 #f;d4 ,robson's ,con/ructive ,lat9 ,ex]cises1 = tea*+ ! ,elem5ts ( ! ,language on a ,sy/em ( ,analysis & ,syn!sis2 ) ,lat9 ,read+ ,lessons & ,copi\s ,vocabul>ies4 ,?ird & ,*eap] ,edition1 ?or\ghly revis$4 #abmo4 #ds4 #f;d4 clo?4 ,robson's ,fir/ ,lat9 ,read+ ,lessons4 ,) ,complete ,vocabul>ies4 ,int5d$ z an ,introduction to ,caes>4 #abmo4 #bs4 #f;d4 clo?4 ,smi?'s ,tacitus2 ,,g]mania1 ,,agricola1 ,,& ,,fir/ ,,book ,,( ,,! ,,annals4 ,) ,engli% ,notes1 orig9al & select$1 & ,btti*]'s rem>ks on ! /yle ( ,tacitus4 ,edit$ by ,dr4 ,,wm4 ,,smi?1 ,editor ( ! ,diction>y ( ,greek & ,roman ,antiquities1 etc4 ,?ird ,edition1 greatly improv$4 #abmo4 #es4 ,caes>4 ,civil ,w>4 ,book ,i4 ,) ,engli% ,notes = ! ,use ( ,/ud5ts prep>+ = ! ,cambridge ,s*ool ,exam9ation4 #abmo4 #as4 #f;d4 ,t]5ce4 ,&ria4 ,) ,engli% ,notes1 ,summ>ies1 & ,life ( ,t]5ce4 ,by ,,new5ham ,,trav]s1 ,b4,a41 ,assi/ant-,ma/] 9 ,univ]sity ,college ,s*ool4 ,fcap4 #hvo4 #cs4 #f;d4 ,,hebrew4 ,hurwitz's ,gramm> ( ! ,hebrew ,language4 ,f\r? ,edition4 #hvo4 #acs4 clo?4 ,or 9 ,two ,p>ts1 sold sep>ately3-- ,,elem5ts4 #ds4 #f;d4 clo?4 ,,etymology ,,& ,,syntax4 #is4 clo?4 ,,fr5*4 ,m]let's ,fr5* ,gramm>4 ,by ,p4 ,f4 ,,m]let1 ,pr(essor ( ,fr5* 9 ,univ]sity ,college1 ,london4 ,new ,edition4 #abmo4 #es4 #f;d4 b\nd4 ,or sold 9 ,two ,p>ts3--,,pronunciation & ,,accid5ce1 #cs4 #f;d42 ,,syntax1 #cs4 #f;d4 7,,key1 #cs4 #f;d47 ,m]let's ,le ,traducteur2 ,selections1 ,hi/orical1 ,dramatic1 & ,miscellane\s1 f ! be/ ,fr5* ,writ]s1 on a plan calculat$ to r5d] read+ & translation peculi>ly s]viceable 9 acquir+ ! ,fr5* ,language2 accompani$ by ,explanatory ,notes1 a ,selection ( ,idioms1 etc4 ,f\rte5? ,edition4 #abmo4 #es4 #f;d4 b\nd4 ,m]let's ,ex]cises on ,fr5* ,composition4 ,consi/+ ( ,extracts f ,engli% ,au?ors to be turn$ into ,fr5*2 ) ,notes indicat+ ! ,diff]5ces 9 ,/yle betwe5 ! two ,languages4 ,a ,li/ ( ,idioms1 ) ,explanations1 ,m]cantile ,t]ms & ,correspond5ce1 ,essays1 etc4 #abmo4 #cs4 #f;d4 ,m]let's ,fr5* ,synonymes1 expla9$ 9 ,alphabetical ,ord]4 ,copi\s ,examples 7f ! 8,diction>y ( ,difficulties074 #abmo4 #bs4 #f;d4 ,m]let's ,ap]u de la ,litt]ature ,franaise4 #abmo4 #bs4 #f;d4 ,m]let's ,/ories f ,fr5* ,writ]s2 9 ,fr5* & ,engli% ,int]l9e> 7f ,m]let's 8,traducteur074 ,second ,edition4 #abmo4 #bs4 cl4 ,lomb>d ,de ,luc's ,classiques ,franais1 l',usage de la ,jeunesse ,prote/ante2 or1 ,selections f ! be/ ,fr5* ,classical ,works1 prec$$ by ,sket*es ( ! ,lives & ,times ( ! ,writ]s4 #abmo4 #cs4 #f;d4 clo?4 ,,italian4 ,smi?'s ,fir/ ,italian ,c\rse2 be+ a ,practical & ,easy ,me?od ( ,le>n+ ! ,elem5ts ( ! ,italian ,language4 ,edit$ f ! ,g]man ( ,,filippi1 aft] ! me?od ( ,dr4 ,,ahn4 #abmo4 #cs4 #f;d4 clo?4 ,,int]l9e> ,,translations4 ,locke's ,sy/em ( ,classical ,in/ruction4 ,,int]l9e> ,,translations #as4 #f;d4 ea*4 ,lat94 ,pha$rus's ,fables ( sop4 ,virgil's neid4 ,book ,i4 ,p>s+ ,lessons to ,virgil4 ,caes>'s ,invasion ( ,brita94 ,greek4 ,lucian's ,dialogues4 ,selections4 ,hom]'s ,iliad4 ,book ,i4 ,x5ophon's ,memorabilia4 ,book ,i4 ,h]odotus's ,hi/ories4 ,selections4 ,fr5*4 ,sismondi2 ! ,battles ( ,cressy & ,poicti]s4 ,g]man4 ,/ories f ,g]man ,writ]s4 ,also1 to accompany ! ,lat9 & ,greek ,s]ies4 ,! ,london ,lat9 ,gramm>4 #abmo4 #as4 #f;d4 ,! ,london ,greek ,gramm>4 #abmo4 #as4 #f;d4 ,,hi/ory1 ,,my?ology1 ,,& ,,antiquities4 ,creasy's 7,pr(essor7 ,hi/ory ( ,engl&4 ,) ,illu/rations4 ,"o ,volume4 ,small #hvo4 ,uni=m ) ,s*mitz's 8,hi/ory ( ,rome10 & ,smi?'s 8,hi/ory ( ,greece40 7,prep>+74 ,s*mitz's ,hi/ory ( ,rome1 f ! ,e>lie/ ,times to ! ,,dea? ,,( ,,commodus1 ,a4,d4 #aib4 ,n9? ,edition4 ,"o ,hundr$ ,engrav+s4 #abmo4 #gs4 #f;d4 clo?4 ,smi?'s ,hi/ory ( ,greece1 f ! ,e>lie/ ,times to ! ,roman ,conque/4 ,) ,supplem5t>y ,*apt]s on ! ,hi/ory ( ,lit]ature & ,art4 ,new ,edition4 ,"o ,hundr$ ,engrav+s on ,wood4 ,l>ge #abmo4 #gs4 #f;d4 clo?4 ,smi?'s ,small] ,hi/ory ( ,greece4 ,) ,illu/rations4 ,fcp4 #hvo4 #cs4 #f;d4 clo?4 ,smi?'s ,diction>y ( ,greek & ,roman ,antiquities4 ,by v>i\s ,writ]s4 ,second ,edition4 ,illu/rat$ by ,sev]al ,hundr$ ,engrav+s on ,wood4 ,"o ?ick volume1 m$ium #hvo4 #b #bs4 clo?4 ,smi?'s ,small] ,diction>y ( ,greek & ,roman ,antiquities4 ,abridg$ f ! l>g] ,diction>y4 ,new ,edition4 ,cr[n #hvo41 #gs4 #f;d4 clo?4 ,smi?'s ,diction>y ( ,greek & ,roman ,biography & ,my?ology4 ,by v>i\s ,writ]s4 ,m$ium #hvo4 ,illu/rat$ by num]\s ,engrav+s on ,wood4 ,complete 9 ,?ree ,volumes4 #hvo4 #e #aes4 #f;d4 clo?4 ,smi?'s ,new ,classical ,diction>y ( ,biography1 ,my?ology1 & ,geography4 ,p>tly bas$ on ! 8,diction>y ( ,greek & ,roman ,biography & ,my?ology40 ,?ird ,edition4 #gej ,illu/rations4 #hvo4 #ahs4 clo?4 ,smi?'s ,small] ,classical ,diction>y ( ,biography1 ,my?ology1 & ,geography4 ,abridg$ f ! l>g] ,diction>y4 ,illu/rat$ by #bjj ,engrav+s on ,wood4 ,new ,edition4 ,cr[n #hvo4 #gs4 #f;d4 clo?4 ,smi?'s ,diction>y ( ,greek & ,roman ,geography4 ,by v>i\s ,writ]s4 ,illu/rat$ ) ,woodcuts ( ,co9s1 ,plans ( ,cities1 etc4 ,two ,volumes #hvo4 #d4 clo?4 ,niebuhr's ,hi/ory ( ,rome4 ,f ! ,e>lie/ ,times to ! ,fir/ ,punic ,w>4 ,f\r? ,edition4 ,translat$ by ,,bi%op ,,?irlwall1 ,,ar*deacon ,,h>e1 ,,dr4 ,,smi?1 & ,,dr4 ,,s*mitz4 ,?ree ,vols4 #hvo4 #a #afs4 ,niebuhr's ,lectures on ! ,hi/ory ( ,rome4 ,f ! ,e>lie/ ,times to ! ,fir/ ,punic ,w>4 ,edit$ by ,,dr4 ,,s*mitz4 ,?ird ,edition4 #hvo4 #hs4 ,newman 7,f4 ,w47 ,! ,odes ( ,horace4 ,translat$ into ,unrhym$ ,metres1 ) ,introduction & ,notes4 ,cr[n #hvo4 #es4 clo?4 ,newman 7,f4 ,w47 ,! ,iliad ( ,hom]1 ,fai?fully translat$ into ,unrhym$ ,metre4 #a vol4 cr[n #hvo4 #fs4 #f;d4 clo?4 ,ak]man's ,numismatic ,manual1 or ,guide to ! ,collection & ,/udy ( ,greek1 ,roman1 & ,engli% ,co9s4 ,_m ,engrav+s4 #hvo4 #a #as4 ,ramsay's 7,george7 ,pr9ciples ( ,psy*ology4 #hvo4 #a,os4 #f;d4 ,,pure ,,ma!matics4 ,de ,morgan's ,elem5ts ( ,ari?metic4 ,fifte5? ,?\s&4 ,royal #abmo4 #es4 clo?4 ,de ,morgan's ,trigonometry & ,d\ble ,algebra4 ,royal #abmo4 #gs4 #f;d4 clo?4 ,ell5b]g]'s ,c\rse ( ,ari?metic1 z taught 9 ! ,pe/alozzian ,s*ool1 ,worksop4 ,po/ #hvo4 #es4 clo?4 ,! ,answ]s to ! ,que/ions 9 ?is ,volume are n[ ready1 price #as4 #f;d4 ,mason's ,fir/ ,book ( ,euclid4 ,expla9$ to ,beg9n]s4 ,fcap4 #hvo4 #as4 #i;d4 ,re9]'s ,lessons on ,=m2 or1 ,an ,introduction to ,geometry1 z giv5 9 a ,pe/alozzian ,s*ool1 ,*eam1 ,surrey4 #abmo4 #cs4 #f;d4 ,re9]'s ,lessons on ,numb]1 z giv5 9 a ,pe/alozzian ,s*ool1 ,*eam1 ,surrey4 ,ma/]'s ,manual1 #es4 ,s*ol>'s ,praxis1 #bs4 ,tables ( ,log>i?ms ,common & ,trigonometrical to ,five ,places4 ,"u ! ,sup]9t5d5ce ( ! ,society = ! ,diffusion ( ,useful ,k4 ,fcap4 #hvo4 #as4 #f;d4 ,f\r ,figure ,log>i?ms & ,anti-,log>i?ms4 ,on a ,c>d4 ,price #as4 ,b>l['s ,tables ( ,squ>es1 ,cubes1 ,squ>e ,roots1 ,cube ,roots1 & ,reciprocals ( all ,integ] ,numb]s1 up to #aj1jjj4 ,royal #abmo4 #hs4 ,,mix$ ,,ma!matics4 ,pott]'s ,treatise on ,me*anics1 = ,junior ,univ]sity ,/ud5ts4 ,by ,,ri*>d ,,pott]1 ,m4,a41 ,pr(essor ( ,natural ,philosophy 9 ,univ]sity ,college1 ,london4 ,?ird ,edition4 #hvo4 #hs4 #f;d4 ,pott]'s ,treatise on ,optics4 ,"p ,i4 ,all ! requisite ,propositions c>ri$ to ,fir/ ,approximations1 ) ! con/ruction ( ,optical ,in/rum5ts1 = ,junior ,univ]sity ,/ud5ts4 ,second ,edition4 #hvo4 #is4 #f;d4 ,pott]'s ,treatise on ,optics4 ,"p ,,ii4 ,! ,high] ,propositions1 ) _! application to ! m p]fect =ms ( ,in/rum5ts4 #hvo4 #abs4 #f;d4 ,pott]'s ,physical ,optics2 or1 ! ,nature & ,prop]ties ( ,light4 ,a ,descriptive & ,exp]im5tal ,treatise4 #ajj ,illu/rations4 #hvo4 #fs4 #f;d4 ,new?'s ,ma!matical ,examples4 ,a graduat$ s]ies ( ,elem5t>y ,examples1 9 ,ari?metic1 ,algebra1 ,log>i?ms1 ,trigonometry1 & ,me*anics4 ,cr[n #hvo4 ,) ,answ]s4 #hs4 #f;d4 clo?4 ,sold also 9 sep>ate ,p>ts1 )\t ,answ]s3-- ,ari?metic1 #bs4 #f;d4 ,algebra1 #bs4 #f;d4 ,trigonometry & ,log>i?ms1 #bs4 #f;d4 ,me*anics1 #bs4 #f;d4 ,new?'s ,elem5ts ( ,me*anics1 includ+ ,hydro/atics1 ) num]\s ,examples4 ,by ,,samuel ,,new?1 ,m4,a41 ,fell[ ( ,univ]sity ,college1 ,london4 ,second ,edition4 ,l>ge #abmo4 #gs4 #f;d4 clo?4 ,new?'s ,fir/ ,book ( ,natural ,philosophy2 or an ,introduction to ! ,/udy ( ,/atics1 ,dynamics1 ,hydro/atics1 & ,optics1 ) num]\s ,examples4 #abmo4 #cs4 #f;d4 clo?4 ,,natural ,,philosophy1 ,,a/ronomy1 ,etc4 ,l>dn]'s ,museum ( ,sci5ce & ,art4 ,complete 9 #ab ,s+le ,volumes1 #ahs41 ornam5tal bo>ds2 or #f ,d\ble ,ones1 #a #as41 cl4 lett]$4 ,also1 h&somely half-b\nd morocco1 #f volumes1 #a #aas4 #f;d4 ,,cont5ts3--,! ,planets2 are !y inhabit$ ,worlds8 ,wea!r ,progno/ics4 ,popul> ,fallacies 9 ,que/ions ( ,physical ,sci5ce4 ,latitudes & ,longitudes4 ,lun> ,influ5ces4 ,meteoric ,/ones & ,%oot+ ,/>s4 ,railway ,accid5ts4 ,light4 ,common ,?+s4--,air4 ,locomotion 9 ! ,unit$ ,/ates4 ,comet>y ,influ5ces4 ,common ,?+s4--,wat]4 ,! ,pott]'s ,art4 ,common ,?+s4--,fire4 ,locomotion & ,transport1 _! ,influ5ce & ,progress4 ,! ,moon4 ,common ,?+s4--,! ,e>?4 ,! ,electric ,telegraph4 ,t]re/rial ,heat4 ,! ,sun4 ,e>?quakes & ,volcanoes4 ,b>omet]1 ,safety ,lamp1 & ,:itwor?'s ,micrometric ,app>atus4 ,/eam4 ,! ,/eam ,eng9e4 ,! ,eye4 ,! ,atmosph]e4 ,"t4 ,common ,?+s4--,pumps4 ,common ,?+s4--,spectacles--,! ,kaleidoscope4 ,clocks & ,wat*es4 ,microscopic ,draw+ & ,engrav+4 ,! ,locomotive4 ,!rmomet]4 ,new ,planets4--,lev]ri] & ,adams's ,planet4 ,magnitude & ,m9ut5ess4 ,common ,?+s4--,! ,almanack4 ,optical ,images4 ,h[ to ,obs]ve ! ,heav5s4 ,common ,?+s4--,! ,look+ ,glass4 ,/ell> ,univ]se4 ,! ,tides4 ,col\r4 ,common ,?+s4--,man4 ,magnify+ ,glasses4 ,in/9ct & ,intellig5ce4 ,! ,sol> ,microscope4 ,! ,cam]a ,lucida4 ,! ,magic ,lant]n4 ,! ,cam]a ,obscura4 ,! ,microscope4 ,! ,:ite ,ants2 _! ,mann]s & ,habits4 ,! ,surface ( ! ,e>?1 or ,fir/ ,notions ( ,geography4 ,sci5ce & ,poetry4 ,! ,bee4 ,/eam ,navigation4 ,electro-,motive ,p[]4 ,?und]1 ,lightn+1 & ! ,aurora ,borealis4 ,! ,pr9t+ ,press4 ,! ,cru/ ( ! ,e>?4 ,comets4 ,! ,/]eoscope4 ,! ,pre-,adamite ,e>?4 ,eclipses4 ,s\nd4 ,l>dn]'s ,animal ,physics1 or ! ,body & its ,functions famili>ly ,expla9$4 #ebj ,illu/rations4 #a vol41 small #hvo4 #abs4 #f;d4 clo?4 ,l>dn]'s ,animal ,physiology = ,s*ools 7*iefly tak5 f ! 8,animal ,physics074 #aij ,illu/rations4 #abmo4 #cs4 #f;d4 clo?4 ,l>dn]'s ,h&-,book ( ,me*anics4 #ceg ,illu/rations4 #a vol41 small #hvo41 #es4 ,l>dn]'s ,h&-,book ( ,hydro/atics1 ,pneumatics1 & ,heat4 #bib ,illu/rations4 #a vol41 small #hvo41 #es4 ,l>dn]'s ,h&-,book ( ,optics4 #bij ,illu/rations4 #a vol41 small #hvo41 #es4 ,l>dn]'s ,h&-,book ( ,electricity1 ,magnetism1 & ,ac\/ics4 #cie ,illu/rations4 #a vol41 small #hvo4 #es4 ,l>dn]'s ,h&-,book ( ,a/ronomy & ,meteorology1 =m+ a companion "w to ! 8,h&-,book ( ,natural ,philosophy40 #cg ,plates1 & upw>ds ( #bjj ,illu/rations on ,wood4 #b vols41 ea* #es41 clo? lett]$4 ,l>dn]'s ,natural ,philosophy = ,s*ools4 #cbh ,illu/rations4 #a vol41 l>ge #abmo41 #cs4 #f;d4 clo?4 ,l>dn]'s ,*emi/ry = ,s*ools4 #agj ,illu/rations4 #a vol41 l>ge #abmo4 #cs4 #f;d4 clo?4 ,pictorial ,illu/rations ( ,sci5ce & ,art4 ,l>ge ,pr9t$ ,%eets1 ea* conta9+ f #ej to #ajj ,engrav$ ,figures4 ,"p ,i4 #as4 #f;d4 #a4 ,me*anic ,p[]s4 #b4 ,ma*9]y4 #c4 ,wat* & ,clock ,"w4 ,"p ,,ii4 #as4 #f;d4 #d4 ,elem5ts ( ,ma*9]y4 #e4 ,motion & ,=ce4 #f4 ,/eam ,eng9e4 ,"p ,,iii4 #as4 #f;d4 #g4 ,hydro/atics4 #h4 ,hydraulics4 #i4 ,pneumatics4 ,l>dn]'s ,popul> ,geology4 7,f 8,! ,museum ( ,sci5ce & ,art407 #bja ,illu/rations4 #bs4 #f;d4 ,l>dn]'s ,common ,?+s ,expla9$4 ,conta9+3 ,air--,e>?--,fire--,wat]--,"t--,! ,almanack--,clocks & ,wat*es--,spectacles--,col\r--,kaleidosc ope--,pumps--,man--,! ,eye--,! ,pr9t+ ,press--,! ,pott]'s ,art--,locomotion & ,transport--,! ,surface ( ! ,e>?1 or ,fir/ ,notions ( ,geography4 7,f 8,! ,museum ( ,sci5ce & ,art407 ,) #bcc ,illu/rations4 ,complete1 #es41 clo? lett]$4 ,sold also 9 ,two ,s]ies1 #bs4 #f;d4 ea*4 ,l>dn]'s ,popul> ,physics4 ,conta9+3 ,magnitude & ,m9ut5ess--,atmosph]e--,?und] & ,lightn+--,t]re/rial ,heat--,meteoric ,/ones--,popul> ,fallacies--,wea!r ,progno/ics--,!rmomet]--,b>omet]--,safet y ,lamp--,:itwor?'s ,micrometric ,app>atus--,electro-,motive ,p[]--,s\nd--,magic ,lant]n--,cam]a ,obscura--,cam]a ,lucida--,look+ ,glass--,/]eoscope--,sci5ce & ,poetry4 7,f 8,! ,museum ( ,sci5ce & ,art407 ,) #he ,illu/rations4 #bs4 #f;d4 clo? lett]$4 ,l>dn]'s ,popul> ,a/ronomy4 ,conta9+3 ,h[ to ,obs]ve ! ,heav5s--,latitudes & ,longitudes --,the,earth--,! ,sun--,! ,moon--,! ,planets3 are !y ,inhabit$8--,! ,new ,planets--,lev]ri] & ,adams's ,planet--,! ,tides--,lun> ,influ5ces--& ! ,/ell> ,univ]se--,light--,comets--,comet>y ,influ5ces--,eclipses--,t]re/rial ,rotation--,lun> ,rotation--,a/ronomical ,in/rum5ts4 7,f 8,! ,museum ( ,sci5ce & ,art407 #ahb ,illu/rations4 ,complete1 #ds4 #f;d4 clo? lett]$4 ,sold also 9 ,two ,s]ies1 #bs4 #f;d4 & #bs4ea*4 ,l>dn] on ! ,microscope4 7,f 8,! ,museum ( ,sci5ce & ,art407 #a vol4 #adg ,engrav+s4 #bs4 ,l>dn] on ! ,bee & ,:ite ,ants2 _! ,mann]s & ,habits2 ) ,illu/rations ( ,animal ,in/9ct & ,intellig5ce4 7,f 8,! ,museum ( ,sci5ce & ,art407 #a vol4 #ace ,illu/rations4 #bs41 clo? lett]$4 ,l>dn] on ,/eam & its ,uses2 includ+ ! ,/eam ,eng9e & ,locomotive1 & ,/eam ,navigation4 7,f 8,! ,museum ( ,sci5ce & ,art407 #a vol41 ) #hi ,illu/rations4 #bs4 ,l>dn] on ! ,electric ,telegraph1 ,popul>is$4 ,) #ajj ,illu/rations4 7,f 8,! ,museum ( ,sci5ce & ,art407 #abmo41 #bej pages4 #bs41 clo? lett]$4 ,! foll[+ ,works f 8,l>dn]'s ,museum ( ,sci5ce & ,art10 may also be _h arrang$ z describ$1 h&somely half b\nd morocco1 clo? sides4 ,common ,?+s4 ,two s]ies 9 "o vol4 #gs4 #f;d4 ,popul> ,a/ronomy4 ,two s]ies 9 "o vol4 #gs4 #j;d4 ,electric ,telegraph1 ) ,/eam & its ,uses4 ,9 "o vol4 #gs4 #j;d4 ,microscope & ,popul> ,physics4 ,9 "o vol4 #gs4 #j;d4 ,popul> ,geology1 & ,bee & ,:ite ,ants4 ,9 "o vol4 #gs4 #f;d4 ,l>dn] on ! ,/eam ,eng9e1 ,/eam ,navigation1 ,roads1 & ,railways4 ,expla9$ & ,illu/rat$4 ,eigh? ,edition4 ,) num]\s ,illu/rations4 #a vol4 l>ge #abmo4 #hs4 #f;d4 ,a ,guide to ! ,/>s = e ,night 9 ! ,ye>4 ,9 ,eight ,planisph]es4 ,) an ,introduction4 #hvo4 #es41 clo?4 ,m9asi's ,me*anical ,diagrams4 ,= ! ,use ( ,lectur]s & ,s*ools4 #ae ,%eets ( ,diagrams1 col\r$1 #aes41 illu/rat+ ! foll[+ subjects3 #a & #b4 ,composition ( ,=ces4--#c4 ,equilibrium4--#d & #e4 ,lev]s4--#f4 ,/eely>d1 ,brady ,balance1 & ,dani% ,balance4--#g4 ,:eel & ,axle4--#h4 ,incl9$ ,plane4--#i1 #aj1 #aa4 ,pulleys4--#ab4 ,hunt]'s ,screw4--#ac & #ad4 ,too!d ,:eels4--#ae4 ,comb9ation ( ! ,me*anical ,p[]s4 ,,logic4 ,de ,morgan's ,=mal ,logic2 or1 ,! ,calculus ( ,inf]5ce1 ,necess>y & ,probable4 #hvo4 #fs4 #f;d4 ,neil's ,art ( ,reason+3 a ,popul> ,exposition ( ! ,pr9ciples ( ,logic1 ,inductive & ,d$uctive2 ) an ,introductory ,outl9e ( ! ,hi/ory ( ,logic1 & an ,app5dix on rec5t ,logical ,developm5ts1 ) ,notes4 ,cr[n #hvo4 #ds4 #f;d41 clo?4 ,,engli% ,,composition4 ,neil's ,elem5ts ( ,rhetoric2 a ,manual ( ! ,laws ( ,ta/e1 includ+ ! ,!ory & ,practice ( ,composition4 ,cr[n #hvo4 #ds4 #f;d41 cl4 ,,draw+4 ,l9eal ,draw+ ,copies = ! e>lie/ ,in/ruction4 ,compris+ upw>ds ( #bjj subjects on #bd %eets1 m\nt$ on #ab pieces ( ?ick pa/ebo>d1 9 a ,portfolio4 ,by ! ,au?or ( 8,draw+ = ,"y ,*ildr540 #es4 #f;d4 ,easy ,draw+ ,copies = ,elem5t>y ,in/ruction4 ,simple ,outl9es )\t ,p]spective4 #fg subjects1 9 a ,portfolio4 ,by ! ,au?or ( 8,draw+ = ,"y ,*ildr540 #fs4 #f;d4 ,sold also 9 ,two ,sets4 ,,set ,i4 ,tw5ty-six ,subjects m\nt$ on ?ick pa/ebo>d1 9 a ,portfolio4 #cs4 #f;d4 ,,set ,,ii4 ,=ty-"o ,subjects m\nt$ on ?ick pa/ebo>d1 9 a ,portfolio4 #cs4 #f;d4 ,! copies are suffici5tly l>ge & bold to be drawn f by =ty or fifty *ildr5 at ! same "t4 ,,s++4 ,a ,musical ,gift f an ,old ,fri5d1 conta9+ ,tw5ty-f\r ,new ,songs = ! ,"y4 ,by ,w4 ,e4 ,,hickson1 au?or ( ! ,moral ,songs ( 8,! ,s++ ,ma/]40 #hvo4 #bs4 #f;d4 ,! ,s++ ,ma/]4 ,conta9+ ,fir/ ,lessons 9 ,s++1 & ! ,notation ( ,music2 ,rudim5ts ( ! ,sci5ce ( ,h>mony2 ,! ,fir/ ,class ,tune ,book2 ,! ,second ,class ,tune ,book2 & ! ,hymn ,tune ,book4 ,six? ,edition4 #hvo4 #fs41 clo? lett]$4 ,sold also 9 ,five ,p>ts1 any ( :i* may be _h sep>ately4 ,i4--,fir/ ,lessons 9 ,s++ & ! ,notation ( ,music4 ,conta9+ ,n9ete5 ,lessons 9 ! ,notation & ,art ( ,read+ ,music1 z adapt$ = ! ,in/ruction ( ,*ildr51 & especially = ,class ,tea*+1 ) ,sixte5 ,vocal ,ex]cises1 arrang$ z simple two-"p h>monies4 #hvo4 #as41 sew$4 ,,ii4--,rudim5ts ( ! ,sci5ce ( ,h>mony or ,?or\gh ,bass4 ,conta9+ a g5]al view ( ! pr9ciples ( ,musical ,composition1 ! ,nature ( ,*ords & ,discords1 mode ( apply+ !m1 & an ,explanation ( ,musical ,t]ms connect$ ) ?is bran* ( ,sci5ce4 #hvo4 #as41 sew$4 ,,iii4--,! ,fir/ ,class ,tune ,book4 ,a ,selection ( ,?irty ,s+le & ,pleas+ ,airs1 arrang$ ) suitable words = "y *ildr54 #hvo4 #as41 sew$4 ,,iv4--,! ,second ,class ,tune ,book4 ,a ,selection ( ,vocal ,music adapt$ = y\? ( diff]5t ages1 & arrang$ 7) suitable words7 z two or ?ree-"p h>monies4 #hvo1 #as4 #f;d4 ,v4--,! ,hymn ,tune ,book4 ,a ,selection ( ,sev5ty popul> ,hymn & ,psalm ,tunes1 arrang$ ) a view ( facilitat+ ! progress ( ,*ildr5 le>n+ to s+ 9 p>ts4 #hvo4 #as4 #f;d4 ,! ,vocal ,ex]cises1 ,moral ,songs1 & ,hymns1 ) ! ,music1 may also be _h1 pr9t$ on ,c>ds1 price ,twop5ce ea* ,c>d1 or ,tw5ty-five = ,?ree ,%ill+s4 ,,*emi/ry4 ,gregory's ,h&-,book ( ,*emi/ry4 ,= ! use ( ,/ud5ts4 ,by ,,william ,,gregory1 ,m4,d41 late ,pr(essor ( ,*emi/ry 9 ! ,univ]sity ( ,ed9burgh4 ,f\r? ,edition1 revis$ & enl>g$4 ,illu/rat$ by ,engrav+s on ,wood4 ,complete 9 ,"o ,volume4 ,l>ge #abmo4 #ahs4 clo?4 ,! ,"w may also be _h 9 two ,volumes1 z "u4 ,,inorganic ,,*emi/ry4 ,f\r? ,edition1 revis$ & enl>g$4 #fs4 #f;d4 clo?4 ,,organic ,,*emi/ry4 ,f\r? ,edition1 v c>efully revis$1 & greatly enl>g$4 #abs41 clo?4 7,sold sep>ately47 ,*emi/ry = ,s*ools4 ,by ,,dr4 ,,l>dn]4 #aij ,illu/rations4 ,l>ge #abmo4 #cs4 #f;d4 clo?4 ,liebig's ,famili> ,lett]s on ,*emi/ry1 9 its ,relations to ,physiology1 ,dietetics1 ,agriculture1 ,comm]ce1 & ,political ,economy4 ,f\r? ,edition1 revis$ & enl>g$1 ) additional ,lett]s4 ,edit$ by ,,dr4 ,,bly?4 ,small #hvo4 #gs4 #f;d4 clo?4 ,liebig's ,lett]s on ,mod]n ,agriculture4 ,small #hvo4 #fs4 ,liebig's ,pr9ciples ( ,agricultural ,*emi/ry2 ) ,special ,ref]5ce to ! late ,rese>*es made 9 ,engl&4 ,small #hvo4 #cs4 #f;d41 clo?4 ,liebig's ,*emi/ry 9 its ,applications to ,agriculture & ,physiology4 ,f\r? ,edition1 revis$4 #hvo4 #fs4 #f;d41 clo?4 ,liebig's ,animal ,*emi/ry2 or1 ,*emi/ry 9 its ,application to ,physiology & ,pa?ology4 ,?ird ,edition4 ,"p ,i4 7! fir/ half ( ! "w74 #hvo4 #fs4 #f;d41 clo?4 ,liebig's ,h&-,book ( ,organic ,analysis2 conta9+ a detail$ ,acc\nt ( ! v>i\s ,me?ods us$ 9 det]m9+ ! ,elem5t>y ,composition ( ,organic ,sub/ances4 ,illu/rat$ by #he ,woodcuts4 #abmo4 #es41 clo?4 ,buns5's ,gasometry2 compris+ ! ,lead+ ,physical & ,*emical ,prop]ties ( ,gases1 toge!r ) ! ,me?ods ( ,gas ,analysis4 ,fifty-n9e ,illu/rations4 #hvo4 #hs4 #f;d41 clo?4 ,whl]'s ,h&-,book ( ,inorganic ,analysis2 ,"o ,hundr$ & ,tw5ty-two ,examples1 illu/rat+ ! mo/ important processes = det]m9+ ! ,elem5t>y composition ( ,m9]al sub/ances4 ,edit$ by ,,dr4 ,a4 ,w4 ,,h(mann1 ,pr(essor 9 ! ,royal ,college ( ,*emi/ry4 ,l>ge #abmo4 ,p>nell on ,dye+ & ,calico ,pr9t+4 7,repr9t$ f ,p>nell's 8,appli$ ,*emi/ry 9 ,manufactures1 ,arts1 & ,dome/ic ,economy1 #ahdd407 ,) ,illu/rations4 #hvo4 #gs41 clo?4 ,,g5]al ,,lit]ature4 ,de ,morgans ,book ( ,almanacs4 ,) an ,index ( ,ref]5ce by :i* ! ,almanac may be f\nd = e ,ye>1 :e!r 9 ,old ,/yle or ,new1 f any ,epo*1 ,anci5t or ,mod]n1 up to ,a4,d4 #bjjj4 ,) means ( f9d+ ! ,day ( ,new or ,full ,moon1 f ,b4,c4 #bjjj to ,a4,d4 #bjjj4 #es41 clo? lett]$4 ,guesses at ,tru?4 ,by ,two ,bro!rs4 ,new ,edition4 ,) an ,index4 ,complete #a vol4 ,small #hvo4 ,h&somely b\nd 9 clo? ) r$ edges4 #ajs4 #f;d4 ,rudall's ,memoir ( ! ,rev4 ,james ,crabb2 late ( ,s\?ampton4 ,) ,portrait4 ,l>ge #abmo41 #fs41 clo?4 ,h]s*ell 7,r4 ,h47 ,! ,jews2 a brief ,sket* ( _! ,pres5t ,/ate & ,future ,expectations4 ,fcap4 #hvo4 #as4 #f;d41 clo?4